Problems on Slope and Y-Intercept

Problems on Slope and Y-Intercept | Slope and Y-Intercept Form Equation Questions and Answers

Problems on Slope and Y-intercept with step-by-step explanations are available here. The equation of the slope-intercept form is y = mx + c. An intercept is a point on the y-axis by which the slope of the line passes. The point where the line crosses the x-axis is known as the x-intercept and the point where the line crosses the y-axis is known as the y-intercept. The Problems on Slope and Y-intercept in coordinate geometry is a combination of different types of questions. Start practicing the problems on slope-intercept from this page and score good marks in the exams.

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  • Worksheet on Slope and Y-intercept

Slope and Y-Intercept Form Problems with Solutions

Example 1.
Determine the slope and y-intercept of the line 6x – 5y + 7 = 0
Solution:
Given that the equation is 6x – 5y – 7 = 0
-5y = -6x + 7
y = −6/−5x + 7/−5.
y = 6/5x – 7/5.
Comparing this with y = mx + c, we have: m = 6/5 and c = -7/5
Therefore, slope = 6/5 and y-intercept = -7/5

Example 2.
What is the slope and y-intercept of the equation 2x – 3y + 4 = 0?
Solution:
Given that the equation is
2x – 3y + 4 = 0
3y = 2x + 4
y = 2/3x + 4/3.
y = 2/3x + 4/3
Comparing the equation with y = mx + c,
we have m = 2/3.
Therefore, the slope of the line is 4/3.

Example 3.
Find the slope and y-intercept of the equation √3x – 3y + 8 = 0
Solution:
Given that the equation is √3x – 3y + 8 = 0
3y = √3x + 8
y = √3/3x + 8/3
Comparing with the equation y = mx + c
Then we have m = √3/3 and c = 8/3
Therefore the slope = √3/3 and y-intercept = 8/3.

Example 4.
What is the slope and y-intercept of the equation 6x – 7y + 8 = 0?
Solution:
Given that the equation is
6x – 7y + 8 = 0
7y = 6x + 8
y = 6/7x + 8/7.
Comparing the equation with y = mx + c,
we have m = 6/7 and c = 8/7
Therefore, the slope of the line is 6/7 and y-intercept = 8/7.

Example 5.
What is the slope and y-intercept of the equation 9x – 10y -11 = 0?
Solution:
Given that the equation is
9x – 10y – 11 = 0
10y = 9x – 11
y = 9/10x – 11/10.
Comparing the equation with y = mx + c,
we have m = 9/10 and c = 11/10
Therefore, the slope of the line is 9/10 and y-intercept = 11/10

Example 6.
If the slope of the line joining the points A(x,2) and B(3,6) is 5/4, find the value of x.
Solution:
Given that the two points are
A(x,2) and B(3,6)
x1 = x, y1 = 2, x2 = 3, y2 = 6
Given slope = 5/4
We know that
x2 – x1/y2 – y1
3 – x/6 – 2 = 5/4
3 – x/4 = 5/4
12 – 4x = 20
12 – 20= 4x
-8 = 4x
x = -2
Hence the value of x = -2.

Example 7.
Find the slope of the line joining the points (4,8) and (5,2)
Solution:
Let A(4,8) and B(5,2) be two points.
Slope of the line = y2 – y1/x2 – x1
= 2-8/5 – 4
= -6/1
= -6/1
= -6
Therefore the slope of the given points are 6.

Example 8.
The following points are plotted in x-y plane.Find the slope and y intercept of the line joining each pair of (1,2) & (3,4)
Solution:
Given that the points are (1,2) and (3,4)
x1 = 1, y1 = 2, x2 = 3 and y2 = 4
slope is (y2-y1)/(x2-x1)
(4 – 2)/(3 -1)
= 2/2
Slope =1
then y=mx+c
you get x-y+1=0
at y axis x=0
y = 1
Therefore y-intercept = 1

Example 9.
What is the slope and y-intercept of the equation x – y + 1 = 0?
Solution:
Given that the equation is
x – y + 1 = 0
y = x + 1
Comparing the equation with y = mx + c,
we have m = 1 and c = 1
Therefore, the slope of the line is 1, and y-intercept = 1.

Example 10.
What is the slope and y-intercept of the equation x – 3y – 7 = 0?
Solution:
Given that the equation is
x – 3y – 7 = 0
3y = x – 7
y = 1/3x – 7/3.
Comparing the equation with y = mx + c,
we have m = 1/3 and c = 7/3
Therefore, the slope of the line is 1/3 and y-intercept = 7/3.

Successor and Predecessor

Successor and Predecessor – Definition, Examples | Difference between the Successor and Predecessor

Successor and Predecessor are the terms that are mentioned just before and after the number or term. Learn the definition, examples, and differences of successor and predecessor. Know the logic to find the successor and predecessor of the number. We are explaining the concept with images and solved examples to make it more clear and easy. Check it out!!

Do Refer: Successor and Predecessor of a Whole Number

Successor and Predecessor in Maths

Check out the What is the Successor and Predecessor from the below details. Also, find out different examples to understand deep about Predecessor and Successor.

What is Predecessor?

The predecessor is the value that comes immediately before/right before the particular value. Suppose that the particular value is x, then the predecessor value is before the value of that particular value i.e., x-1. Therefore, to find the predecessor value of any number, we have to subtract 1 from the given value.

Examples:

If x = 15, then the predecessor value of 15 is 15 – 1 = 14
If x = 21, then the predecessor value of 21 is 21 – 1 = 20
If x = 49, then the predecessor value of 49 is 49 – 1 = 48
If x = 90, then the predecessor value of 90 is 90 – 1 = 90
If x = 115, then the predecessor value of 115 is 115 – 1 = 114

Therefore, the predecessor of any number is one less than the original whole number.

What is Successor?

The successor is the value that comes immediately after/right after the particular value. Suppose that the particular value is x, then the successor value is after the value of that particular value i.e., x + 1. Therefore, to find the successor value of any number, we have to add 1 to the given value.

Examples:
If x = 15, then the successor value of 15 is 15 + 1 = 16
If x = 21, then the successor value of 21 is 21 + 1 = 22
If x = 49, then the successor value of 49 is 49 + 1 = 50
If x = 90, then the successor value of 90 is 90 + 1 = 91
If x = 115, then the successor value of 115 is 115 + 1 = 116

Therefore, the successor of any number is one greater than the original whole number.

How to find the Successor and Predecessor of a Number?

To find the predecessor and successor of any value, we apply the basic subtraction and addition methods.

To evaluate the successor and predecessor of any value, we have to apply the basic method of addition and subtraction, respectively. For the successor, we need to add 1 to the given number whereas for the predecessor we have to subtract 1 from the given number. Finding a successor and predecessor is very easy and quick.

  • Successor = Given number + 1
  • Predecessor = Given number – 1

Let us see some solved examples here to understand better.

Successor and Predecessor Examples

Example 1.
Find the successor of the following numbers:
(i) 15
(ii) -11
(iii) -85
(iv) 91
(v) 149
(vi) 44
(vii) 87
(viii) 78

Solution:
The successor values of the numbers are as follows:
(i) 15 + 1 = 16
The successor value of 15 is 16
(ii) -11 + 1 = -10
The successor value of -11 is – 10
(iii) -85 + 1 = -84
The successor value of -85 is -84
(iv) 91 + 1 = 92
The successor value of 91 is 92
(v) 149 + 1 = 150
The successor value of 149 is 150
(vi) 44 + 1 = 45
The successor value of 44 is 45
(vii) 87 + 1 = 88
The successor value of 87 is 88
(viii) 78 + 1 = 79
The successor value of 78 is 79

Example 2.
Find the predecessor of the following numbers:
(i) -15
(ii) -81
(iii) 65
(iv) -9
(v) 22
(vi) 198
(vii) 55

Solution:
The predecessor values of the numbers are as follows:
(i) -15 – 1 = -16
The predecessor value of -15 is -16
(ii) -81 – 1 = -82
The predecessor value of -81 is -82
(iii) 65 – 1 = 64
The predecessor value of 65 is 64
(iv) – 9 – 1 = -10
The predecessor value of -9 is -10
(v) 22 – 1 = 21
The predecessor value of 22 is 21
(vi) 198 – 1 = 197
The predecessor value of 198 is 197
(vii) 55 – 1 = 54
The predecessor value of 55 is 54

Example 3.
Write the successor and predecessor of the following numbers:
(i) 94
(ii) 114
(iii) 32
(iv) 65
(v) 78

Solution:
The successor and predecessor of the numbers are as follows:
(i) The number is 94
Successor value of 94 is 94 + 1 = 95
Predecessor value of 94 is 94 – 1 = 93
(ii) The number is 114
Successor value of 114 is 114 + 1 = 115
Predecessor value of 114 is 114 – 1 = 113
(iii)The number is 32
Successor value of 32 is 32 + 1 = 33
Predecessor value of 32 is 32 – 1 = 31
(iv)The number is 65
Successor value of 65 is 65 + 1 = 66
Predecessor value of 65 is 65 – 1 = 64
(v)The number is 78
Successor value of 78 is 78 + 1 = 79
Predecessor value of 78 is 78 – 1 = 77

FAQs on Successor and Predecessor

1. What is the main difference between successor and predecessor

The successor is the number that comes after the original number. The predecessor is the number that comes before the original number.

2. Is there any natural number available that has no predecessor value?

Yes, there is a natural number that has no predecessor value i.e., 1. The reason for this is that natural numbers start from 1.

3. Is there any natural number available that has no successor value?

No, there is no natural number that has no successor value. As the natural numbers are infinite and it has no last value.

4. What is the formula for successor and predecessor numbers?

The formula for successor and predecessor are

  • Successor = Original number + 1
  • Predecessor = Original number – 1
Quadrants and Convention for Signs of Coordinates

Quadrants and Convention for Signs of Coordinates | What are 4 Quadrants in Order?

Are you searching for the information regarding the Quadrants and Convention for Signs of Coordinates on various websites? If our guess is correct, then you are on the right page. Stop your search and start your preparation now. Here you can get complete and brief info about the quadrants and signs of coordinates. Know what is quadrant and what are the signs of the coordinates in each quadrant from here. Also, the students of grade 9 can find examples on the quadrants of coordinate geometry in the below sections.

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What are the Sign Convention of the Coordinates?

What is Quadrant?
In a coordinate plane, a quadrant is defined as the region of a cartesian plane formed by the two axes i.e, x-axis and y-axis. The region of the plane falling in XOY is the first quadrant, the region of the plane falling in X’OY is the second quadrant, the region of the plane falling in X’OY’ is the third quadrant, and the region of the plane falling in XOY’ is the fourth quadrant.

Four Quadrants in a Plane:
1st Quadrant: The upper right-hand corner is the first quadrant. In this quadrant x and y values are positive.
2nd Quadrant: The upper left-hand corner is the second quadrant. In this quadrant, the value of x is negative and the value of y is positive.
3rd Quadrant: The lower left-hand corner is the third quadrant. In this quadrant both the x and y values are negative.
4th Quadrant: The fourth quarter is at the lower right-hand corner, in which the x value is positive(+) and y value is negative(-).

Quadrant

What are the Signs of the Coordinates in Each Quadrant?

Quadrants x-coordinate y-coordinate
Quadrant – I Positive (+) Positive (+)
Quadrant – II Negative (-) Positive (+)
Quadrant – III Negative (-) Negative (-)
Quadrant – IV Positive (+) Negative (-)

From the table, we observe that there are two signs of the coordinates in each quadrant.
Quadrant – I: In the first quadrant both the coordinates have positive signs.
Quadrant – II: In the second quadrant x-coordinate has a negative sign and the y-coordinate has a positive sign.
Quadrant – III: In the third quadrant both the coordinates have negative signs.
Quadrant – IV: In the fourth quadrant x-coordinate has a positive sign and the y-coordinate has a negative sign.

Quadrants and Convention for Signs of Coordinates Examples

Example 1.
Check and answer to which quadrants do the given points lie?
(2, 4)
(-4, -5)
(-3, 6)
(4, -4)
Solution:
The coordinates of (2, 4) are positive, thus it lies on the first quadrant.
Quadrant img_1
The coordinates of the point (-4, -5) are both negative, thus it lies on the third quadrant.
Quadrant img_2
The coordinates of the point (-3, 6) are negative on the x-axis and positive on the y-axis, thus it lies on the second quadrant.
Quadrant img_3
The coordinates of the point (4, -4) are positive on the x-axis and negative on the y-axis, thus it lies on the fourth quadrant.
Quadrant img_4

Example 2.
Give an example of a point that lies on the third quadrant.
Solution:
In the third quadrant, the coordinates of the x-axis and y-axis both are negative.
(-6, -8) is an example of a point in the third quadrant.

Example 3.
Where does the point (-6, 0) lie in the cartesian plane? Does it lie in a quadrant?
Solution:
The point (-6, 0) lies on the horizontal axis or x-axis,
which is at a distance of 6 units from the origin. It does not lie in any quadrant.
Quadrant img_5

Example 4.
Give an example of a point that lies on the second quadrant.
Solution:
In the third quadrant, the coordinates of the x-axis are negative and the y-axis is positive.
(-4, 7) is an example of a point in the third quadrant.

Example 5.
Where does the point (-1,4) lie?
Solution:
Given that the point is (-1,4)
The point x coordinate is negative and the y coordinate is positive so the point lies in the second quadrant.
Quadrant img_6

FAQs on Quadrants and Convention for Signs of Coordinates

1. What is a quadrant?

A quadrant is defined as the region of a cartesian plane. It is formed when the x-axis and y-axis both intersect each other.

2. What is the intersection of two axes called?

The intersection of the x-axis and y-axis is called the reference point or the origin.

3. What are the Four Quadrants in a Coordinate Plane?

The graph has been divided into 4 parts.
1st Quadrant: In a graph, the upper right-hand corner is the first quadrant. In this quadrant x and y values are positive.
2nd Quadrant: In a graph, the upper left-hand corner is the second quadrant. In this quadrant, the value of x is negative and the value of y is positive.
3rd Quadrant: In the graph, the lower left-hand corner is the third quadrant. In this quadrant both the x and y values are negative.
4th Quadrant: In the fourth quarter is at the lower right-hand corner, which the x value is positive and the y value is negative.

Position of a Point in a Plane

Position of a Point in a Plane | How do you find the Position of a Point on a Plane?

The position of a point in a plane is provided by the coordinates on the graph. XOX’ and YOY’ are the two intersecting and perpendicular lines. We know that x-axis and y-axis divide the XY-plane into four parts known as quadrants. The position of a point in the plane is defined with the help of ordered pairs. Let us discuss more about the position of a point in a plane from this article. Scroll down this page to practice the examples problems on the position of a point in the plane from the chapter reflection.

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Position of a Point in a Plane

The position of a point in the plane is expressed by a set of values or numbers called coordinates. There are two coordinates in the geometry they are x-coordinate known as abscissa and y-coordinate known as ordinate. Any point in a plane is referred to by a point (x, y), where the x-value is the position of the point with reference to the x-axis and the y-value is the position of the point with reference to the y-axis.

Position of a Point in a Plane Examples

Check out the problems given in the below section to understand in depth about the position of a point in a plane.

Example 1.
Find the position of a point in a plane (2,4).
Solution:
Given that the point is (2,4)
Consider the point (2,4) has A.
Plot the point A(2,4) in the first quadrant takes a distance of 2 from the x-axis and 4 from the y axis.
So, the coordinates of A are (2, 4). We express it by writing A = (2, 4).
And also consider B, C, and D are also points in the other quadrant whose distance from the y-axis and the x-axis are the same point. But due to their positions in different quadrants, their coordinates are different. Thus, the coordinates of B, in the second quadrant, are (-2, 4), and C is the third quadrant, are (-2, -4) and D is the fourth quadrant, are (2, -4).
Position of a point in a plane_1

Example 2.
Find the position of a point in a plane (6,7).
Solution:
Given that the point is (6,7)
Consider the point (6,7) has A.
Plot the point A(6,7) in the first quadrant takes a distance of 6 from the x-axis and 7 from the y axis.
So, the coordinates of A are (6, 7). We express it by writing A = (6, 7).
And also consider B, C, and D are also points in the other quadrant whose distance from the y-axis and the x-axis are the same point. But due to their positions in different quadrants, their coordinates are different. Thus, the coordinates of B, in the second quadrant, are (-6, 7), and C is the third quadrant, are (-6, -7) and D is the fourth quadrant, are (6, -7).
Position of a point in a plane_2

Example 3.
Find the position of a point in a plane (12,14).
Solution:
Given that the point is (12,14)
Consider the point (12,14) has A.
Plot the point A(12,14) in the first quadrant takes a distance of 12 from the x-axis and 14 from the y axis.
So, the coordinates of A are (12, 14). We express it by writing A = (12, 14).
And also consider B, C, and D are also points in the other quadrant whose distance from the y-axis and the x-axis are the same point. But due to their positions in different quadrants, their coordinates are different. Thus, the coordinates of B, in the second quadrant, are (-12, 14), and C is the third quadrant, are (-12, -14) and D is the fourth quadrant, are (12, -14)
Position of a point in a plane_3

Example 4.
Find the position of a point in a plane (5,10).
Solution:
Given that the point is (5,10)
Consider the point (5,10) has A.
Plot the point A(5,10) in the first quadrant takes a distance of 5 from the x-axis and 10 from the y axis.
So, the coordinates of A are (5, 10). We express it by writing A = (5, 10).
And also consider B, C, and D are also points in the other quadrant whose distance from the y-axis and the x-axis are the same point. But due to their positions in different quadrants, their coordinates are different. Thus, the coordinates of B, in the second quadrant, are (-5, 10), and C is the third quadrant, are (-5, -10) and D is the fourth quadrant, are (5, -10).
Position of a point in a plane_4

Example 5.
Find the position of a point in a plane (2,9).
Solution:
Given that the point is (2,9)
Consider the point (2,9) has A.
Plot the point A(2,9) in the first quadrant takes a distance of 2 from the x axis and 9 from the y axis.
So, the coordinates of A are (2, 9). We express it by writing A = (2, 9).
And also consider B, C, and D are also points in the other quadrant whose distance from the y-axis and the x-axis are the same point. But due to their positions in different quadrants, their coordinates are different. Thus, the coordinates of B, in the second quadrant, are (-2, 9), and C is the third quadrant, are (-2, -9) and D is the fourth quadrant, are (2, -9).
Position of a point in a plane_5

FAQs on Position of a Point in a Plane

1. What is the position of a plane?

A plane is a surface in which any two points are taken and the straight line drawn to join these two points that lie within that plane or surface.

2. What is the first coordinate of a point?

If the coordinates points are (a,b). The first point named in the ordered pair is called x-coordinate and the second is called the y-coordinate.

3. How many points determine a line?

If any two distinct points in a plane determine a line, and the equation is also determined by the coordinates of the points.

Coordinates of a Point

Coordinates of a Point – Definition, Meaning, Examples | How to Find the Coordinates of a point?

The complete information on finding the coordinates of a point are given on this page. So, go through the entire article to learn the points on the coordinate plane with a detailed explanation. Also, learn the basics of coordinate geometry from here. We suggest you practice the problems given at the end of this page to know how much knowledge you have gained from this topic. So, that you can know where you are lagging and overcome the difficulties from the chapter coordinate geometry.

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What do you mean by Coordinates of a Point?

The exact position of the points on the cartesian plane is known as the coordinates of the points. We know that there are two coordinates on the plane one is x-coordinate and the other is y-coordinate. These coordinates represent the exact position of the point on a coordinate plane. There are four quadrants in the coordinate plane first quadrant has both positive values, the second quadrant has a negative x-axis, positive y-axis, the third quadrant has both negative signs and the fourth quadrant has a positive x-axis and negative y-axis.
Coordinates of a point

Coordinates Meaning

A coordinate is a set of values that defines the accurate location of the points on a plane. Each point in a plane is assigned with some set of numbers called coordinates of a plane. There are two coordinates in the coordinate plane the first coordinate of a point is called x-coordinate and the second coordinate of a point is called y-coordinate.
x-coordinate: 
The first number in the ordered pair is known as the x-coordinate of the point. This represents the points on the x-axis. The values on the x-axis may be positive and negative. The positive values on the x-axis will be the right side of the origin and the negative values on the x-axis will be the left side of the origin.
y-coordinate: 
The second number in the ordered pair is known as the y-coordinate of the two-dimensional plane. The numbers on the y-axis are both positive and negative. The positive values on the y-axis will be located on the north and the negative values on the y-axis will be located on the south part of the graph.
x-axis: The horizontal line on the coordinate plane is known as the x-axis. The positive values on the horizontal line are indicated as the x-axis and the negative values on the horizontal line are indicated as the x’-axis.
y-axis: The vertical line on the coordinate plane is known as the y-axis. The positive values on the vertical line are indicated as the y-axis and the negative values on the vertical line are indicated as the y’-axis.

How to find Coordinates of a Point?

There are some necessary steps to follow in order to find the coordinates of a point.
1. XOX’ and YOY’ are the co-ordinate axes on the graph.
2. To locate the point on the axes we need to draw a perpendicular from P on X’OX.
3. P(OT, PT) coordinate of a point is formed.

Coordinates of a Point Examples

Example 1.
To find the coordinate of point P the distance of P from the x-axis is 4 units and the distance of P from the y axis is 6 units.
Solution:
Therefore the coordinates of a point P are (4,6).
Here a = 4 locates on x coordinate
b = 6 locates on y coordinates
Coordinates of a point_1

Example 2.
To find the coordinate of point P the distance of P from the y-axis is 7 units and the distance of P from the x-axis is 7 units.
Solution:
Therefore the coordinates of a point P are (7,7).
Here a = 7 locates on x coordinate
b = 7 locates on y coordinates
Coordinates of a point_2

Example 3.
In the adjoining figure, XOX’ and YOY’ are the coordinate axis. Find out the coordinates of point A?
Solution:
To locate the position of point A, draw a straight line and perpendicular line from point A to the x-axis OX, y-axis OY’.
Measure the distance between the point on the x-axis and origin, a new point on the y-axis, and origin.
The value of that point on the x-axis is 3. And the value of the point on the y-axis is -5.
So, the x coordinate is 4, y coordinate is -5.
Therefore The ordered pair is A (4, -5)
Coordinates of a point_3

Example 4.
Find the coordinates of three marks in the following figure?
Solution:
To locate the position of point A
The ordered pair A is located in the first quadrant where both coordinates are positive.
The perpendicular distance of A from the x-axis is 4 units and the y-axis is 4 units.
So, the abscissa is 4, ordinate is 4
Therefore, the coordinates of P are (4, 4).
To locate the position of point B
The ordered pair B is also located in quadrant 1.
The perpendicular of point B from the x-axis is 0.
So, the x coordinate is 0 and the point lies on the y-axis.
The distance of B from the origin on the y-axis is 3 units.
Therefore the y coordinate is 3.
The coordinates of point Q
B (0, 3)
To locate the position of point C
Point C is located on the x-axis means its y coordinate is 0.
The distance of point C from the origin is -3 units.
So the x coordinate is -3.
Thus the Coordinates of point C (0, -3).

Example 5.
To find the coordinate of point P the distance of P from the y-axis is 3 units and the distance of P from the x-axis is -8 units.
Solution:
Therefore the coordinates of a point P are (3,-8).
Here a = 3 locates on x coordinate
b = -8 locates on y coordinates.
Coordinates of a point_4

FAQs on Coordinates of a Point

1. How many coordinates are there at a point?

The point in a two-dimensional plane has two coordinates. One is x coordinate and it is also called abscissa and the second is y coordinate and it is also called ordinate.

2. What is meant by coordinates?

In a plane, the set of values of a point are called coordinates.

3. What do you mean by the coordinates of a point?

A pair of numbers describes the position of a point in a coordinate plane by using the horizontal and vertical distances from the two reference axis they represent by (x,y) means x coordinate and y coordinate.

Independent Variables and Dependent Variables

Independent Variables and Dependent Variables – Definition, Examples | How do you Identify Independent and Dependent Variables?

Independent Variables and Dependent Variables are the categories of the variables in maths. Before going into the concept of Independent and Dependent Variables in coordinate geometry we will discuss variables in maths. Also, let us know the similarities of independent variables and dependent variables, differences between Independent Variables and Dependent Variables with example problems. Hence utilize the content provided in this article and score the highest marks in the exams.

Variable – Definition

In maths, a variable is an alphabetic character that expresses the value or number. A variable is used to represent the unknown number of unknown values. A variable is a value that can vary the expression or equation depending on the conditions.

Independent Variables and Dependent Variables

Depending and Independent Variables are the variables in maths. A cartesian plane or coordinate system is used for plotting the chain of points that shows the relationship between the independent variables and dependent variables. the x-axis shows the independent variables i.e, input values and the y-axis shows the output values of the dependent variables.

Compare and Contrast Independent Variables and Dependent Variables?

In coordinate geometry, the independent variables can be seen on the x-axis and comprise the input values. In short, we can say that the independent variable is an arbitrary input on the cartesian plane. Whereas the dependent variable is an arbitrary output on the cartesian plane i.e., on the y-axis. The dependent variables are found on the y-axis and include output values.

Examples of Independent Variables and Dependent Variables

Check out the below examples to know how to solve the problems on dependent variables and independent variables.

Example 1.
Find the independent variables of the equation 3y = 6x + 1
Solution:
Given that
3y = 6x + 1 is an algebraic expression
Here x is an independent variable.
Each value of x will be a different value of y.

Example 2.
Find the dependent variables of the equation y = 4x.
Solution:
Given that
y = 4x is an algebraic expression
Here y depends on x
So, if x = 1 then y = 4.

Example 3.
Find the variables of the equation 2x + 6 = 20
Solution:
Given that
2x + 6 = 20 is an algebraic expression
Here x is a variable
2 is the coefficient
6 and 20 are the constants.

Example 4.
Find the independent variables of the equation 4y = 5x + 3
Solution:
Given that
4y = 5x + 3 is an algebraic expression
Here x is an independent variable.
Each value of x will be a different value of y.

Example 5.
Find the dependent variables of the equation y = 7x.
Solution:
Given that
y = 7x is an algebraic expression
Here y depends on x
So, if x = 1 then y = 7.

FAQs on Correlation between Independent Variables and Dependent Variables

1. What is meant by Dependent Variables?

The dependent variable is a variable whose quality depends on another variable in its condition. If the estimation of the word variable is dependably said to be dependent on the free variable of math conditions.

2. What is meant by Independent Variables?

In an algebraic equation, an independent variable is related to a variable whose values are independent of changes. If x and y are two variables of an equation and every value of x is linked with any other value of y, then ‘y’ value is said to be a function of x value, it is known as an independent variable, and therefore ‘y’ value is known as a dependent variable.

3. How do you write the independent and dependent variables in an equation?

y = a + bx
Here a and b are constant numbers.
x is the independent variable, and
y is the dependent variable. Therefore we choose a value to substitute for the independent variable and then solve for the dependent variable.

Worksheet on Irrational Numbers

Worksheet on Irrational Numbers | Practice Problems on Irrational Numbers

In the number system, irrational numbers are numbers that can’t be written in fractional form. The different types of problems related to irrational numbers are comparison, representation on the number line, rationalizing the denominator and so on. All these types of irrational numbers questions are covered in the Worksheet on Irrational Numbers. So interested students have a look at them and start solving for scoring better marks in the exam.

The problems related to rationalizing the denominator helps to solve the functions of irrational numbers. Given some of the questions are involving calculations of irrational numbers.

More Related Articles:

Irrational Numbers Worksheet

Question 1:
Check if the below numbers are rational or irrational.
π, \(\frac { 1 }{ √2 } \), \(\frac { √3 }{ 8 } \), √5

Solution:

Since the irrational numbers non-repeating or non-terminating.
So, π, \(\frac { 1 }{ √2 } \), \(\frac { √3 }{ 8 } \), √5 are irrational numbers.


Question 2:
Determine whether the following numbers are rational or irrational.
\(\frac { 1 }{ 2 } \), 8, \(\frac { 15 }{ 6 } \), \(\frac { 76 }{ 5 } \)

Solution:

Since the decimal expansion of a rational number either repeats or terminates.
So, \(\frac { 1 }{ 2 } \), 8, \(\frac { 15 }{ 6 } \), \(\frac { 76 }{ 5 } \) are rational numbers.


Question 3:
Compare √6 and √5

Solution:

Given two irrational numbers are √6 and √5
We know that if ‘p’ and ‘q’ are two numbers such that ‘q’ is greater than ‘p’, then q² will be greater than p². So, square the given numbers.
√6 = √6 x √6 = (√6)² = 6
√5 = √5 x √5 = (√5)² = 5
6 is greater than 5.
So, √6 is greater than √5.


Question 4:
Arrange the following irrational numbers in descending order.
√17, √3, √21, √10, √51

Solution:

Given irrational numbers are √17, √3, √21, √10, √51
We know that if ‘p’ and ‘q’ are two numbers such that ‘q’ is greater than ‘p’, then q² will be greater than p². So, square the given numbers.
√17 = √17 x √17 = (√17)² = 17
√3 = √3 x √3 = (√3)² = 3
√21 = √21 x √21 = (√21)² = 21
√10 = √10 x √10 = (√10)² = 10
√51 = √51 x √51 = (√51)² = 51
Arranging the descending order means placing the numbers from the greatest to the smallest.
So, 51 > 21 > 17 > 10 > 3
Hence, the descending order of numbers is √51 > √21 > √17 > √10 > √3.


Question 5:
Write the irrational numbers ∜6, √3 and ∛7 in ascending and descending orders.

Solution:

Given irrational numbers are ∜6, √3 and ∛7
Order of the irrational numbers are 4, 2, 3
The least common multiple of (4, 2, 3) = 12
So, we have to change the order of each number as 12
Change ∜5 as 12th root
∜6 = (4 x 3) √6³
= 12 √216
√3 = (2 x 6) √36
= 12 √729
∛7 = (3 x 4) √74
= 12 √2401
216 < 729 < 2401.
Therefore, ascending order is ∜6, √3, ∛7 and descending order is ∛7, √3 and ∜6.


Question 6:
Find an irrational number between √6 and 6.

Solution:

Given two real numbers are √6 and 6
A real number between √6 and 6 is \(\frac { √6 + 6 }{ 2 } \) = ½√6 + 1
But 1 is a rational number and ½√6 is an irrational number. The sum of a rational number and an irrational number is irrational.
So, ½√6 + 1 is an irrational number that lies between √6 and 6.


Question 7:
Insert two irrational numbers between √13 and √19.

Solution:

Given two real numbers are √13 and √19
Consider the squares of √13 and √19
(√13)² = 13
(√19)² = 19
Since the numbers 14, 17 lie between 13 and 19 i.e between (√13)² and (√19)²
Therefore, √13 and √19 are √14 and √17.
Hence two irrational numbers between √13 and √19 are √14 and √17.


Question 8:
Rationalize \(\frac { 12 }{ 3√10 } \)

Solution:

Given fraction is \(\frac { 12 }{ 3√10 } \)
Since the given fraction has an irrational denominator, so we need to rationalize this and make it more simple. So, to rationalize this, we will multiply the numerator and denominator of the given fraction by root 10, i.e., √10. So,
\(\frac { 12 }{ 3√10 } \) = \(\frac { 12 }{ 3√10 } \) x \(\frac { √10 }{ √10 } \)
= \(\frac { 12√10 }{ 3(10) } \)
= \(\frac { 12√10 }{ 30 } \)
= \(\frac { 2√10 }{ 5 } \)
So, the required rationalized form is \(\frac { 2√10 }{ 5 } \)


Question 9:
Rationalize \(\frac { 25 }{ 16 + 2√31 } \)

Solution:

Given fraction is \(\frac { 25 }{ 16 + 2√31 } \)
Multiply numeration, denominator by (16 – 2√31)
\(\frac { 25 }{ 16 + 2√31 } \) = \(\frac { 25 }{ 16 + 2√31 } \) x \(\frac { 16 – 2√31 }{ 16 – 2√31 } \)  [(a + b)(a – b) = a² – b²]
= \(\frac { 25(16 – 2√31) }{ 16² – (2√31)² } \)
= \(\frac { 400 – 50√31 }{ 256 – 124 } \)
= \(\frac { 400 – 50√31 }{ 132 } \)
= \(\frac { 200 – 25√31 }{ 66 } \)
So, the required rationalized form is \(\frac { 200 – 25√31 }{ 66 } \).


Question 10:
Rationalize \(\frac { 8 + √6 }{ √5 – 2√7 } \)

Solution:

Given fraction is \(\frac { 8 + √6 }{ √5 + 2√7 } \)
Since, the given problem has an irrational term in the denominator with addition format. So we need to rationalize using the method of multiplication by the conjugate. So,
\(\frac { 8 + √6 }{ √5 + 2√7 } \) = \(\frac { 8 + √6 }{ √5 + 2√7 } \) x \(\frac { √5 – 2√7 }{ √5 – 2√7 } \)
= \(\frac { (8 + √6)(√5 – 2√7) }{ 5 – 4(7) } \)
= \(\frac { 8√5 – 16√7 + √30 – 2√35 }{ 5 – 28 } \)
= \(\frac { 8√5 – 16√7 + √30 – 2√35 }{ -23 } \)
So, the required rationalized form is \(\frac { -8√5 + 16√7 – √30 + 2√35 }{ 23 } \).


Reflection of a Point in the y-axis

Reflection of a Point in the y-axis – Definition, Meaning, Rules | How do you find the Reflection of a Point on the y-axis?

In Maths the word reflection is used in graphs in coordinate geometry. When the reflection of a point in the y-axis, the sign of the y-coordinate remains the same and the sign of the x-coordinate will be varied. The graph is a reflection with respect to the y-axis that is x = 0.

Also Practice:

Reflection of a Point in the y-axis – Definition & Meaning

When the point is reflected in the y-axis, then the x-coordinate will be changed and the y-axis remains the same. For example, the point P(x, y) is P(-x, y).
Reflection of a Point in the y-axis

What are the Rules to Find the Reflection of a Point in the y-axis?

The rules to follow the reflection of a point in the y-axis are shown below.
i. Change the sign of the x-coordinate (abscissa).
ii. Maintain the y-coordinate.

Reflection of a Point in the y-axis Examples

Example 1.
Write the coordinates of the image of the point (-1, 2) when reflected in the y-axis.
Solution:
Given that the point is (-1, 2)
As per the rule of reflection of points in the y-axis, the x coordinate will become negative of its value while the y-coordinate will stay the same.
The image of (-1, 2) is (1, 2).
Reflection of the point in the y-axis_1

Example 2.
Write the coordinates of the image of the point (3, -4) when reflected in the y-axis.
Solution:
Given that the point is (3, -4)
As per the rule of reflection of points in the y-axis, the x coordinate will become negative of its value while the y-coordinate will stay the same.
The image of (3, -4) is (-3, -4).
Reflection of the point in the y-axis_2

Example 3.
Write the coordinates of the image of the point (-5, 6) when reflected in the y-axis.
Solution:
Given that the point is (-5, 6)
As per the rule of reflection of points in the y-axis, the x coordinate will become negative of its value while the y-coordinate will stay the same.
The image of (-5, 6) is (5, 6).
Reflection of the point in the y-axis_3

Example 4.
Write the coordinates of the image of the point (7, -8) when reflected in the y-axis.
Solution:
Given that the point is (7,-8)
As per the rule of reflection of points in the y-axis, the x coordinate will become negative of its value while the y-coordinate will stay the same.
The image of (7, -8) is (-7,-8).
Reflection of the point in the y-axis_4

Example 5.
Write the coordinates of the image of the point (-9, 10) when reflected in the y-axis.
Solution:
Given that the point is (-9, 10)
As per the rule of reflection of points in the y-axis, the x coordinate will become negative of its value while the y-coordinate will stay the same.
The image of (-9, 10) is (9, 10).
Reflection of the point in the y-axis_5

FAQs on Reflection of a Point in the y-axis

1. What does the point that lies on the y-axis represent?

If a point lies on an axis, one of its coordinates must be zero. If point A looks at how far the point is from the origin along the x-axis, the answer is zero. Therefore, the x-coordinate is zero. Any point that lies on the y-axis has an x-coordinate of zero.

2. What is the distance of a point from the y-axis called?

The distance of a point from the y – axis is called x-coordinate, or abscissa, and the distance of the point from the x-axis is called y-coordinate, or ordinate.

3. What is the rule of reflection?

To perform a reflection, we need a line of reflection; the resulting orientation of the two figures is opposite. Corresponding parts of the figures are the same distance from the line of reflection. Ordered pair rules reflect over the x-axis: (x, -y), y-axis: (-x, y), line y=x: (y, x).

Section Formula

Section Formula in Coordinate Geometry | Internal & External Division Section Formula

Are you browsing various sites to learn about section formulae and distance formulae? Then have a look at this article to know briefly about the section formula and the formulas used with examples. Usually, the section formulae help to find the coordinates of the point to divide line joining points in a ratio. Scroll down this page to find the derivation of the section formula.

Also Read:

Section Formula

In geometry, the section formula is used to find the ratio in which the line segment is divided by point externally or internally. We use a section formula to find the incenter, centroid, and excenter of the triangle. We can find the midpoint of the line segment with the help of the section formula.

Section Formula Derivation

section-formula-proof
Let P(x1, y1) and Q(x2, y2) be two points in the above XY-plane.
Let M(x, y) divide the line segment PQ equally with the ratio m:n.
PA, MN and QR are drawn parallel to x-axis.
∠MPS = ∠QMB (corresponding angles)
∠MSP = ∠QBM = 90°
By following AA similarity
ΔPMS ~ ΔMQB
PM/MQ = PS/MB = MS/QB = m/n
PS = AN = ON – OA = x – x1
MB = NR = OR – ON = x2 – x
MS = MN – SN = y – y1
QB = RQ – RB = y2 – y
m/n = x – x1/x2 – x = y – y1/y2 – y
x = mx2 + nx1/m + n
m/n = y – y1/y2 – y
y = my2 + ny1/m + n
Section Formula (Internally):
m:n = (mx2 + nx1/m + n, my2 + ny1/m + n)
Section Formula (Externally):
m:n = (mx2 – nx1/m – n, my2 – ny1/m – n)

Section Formula Examples

Example 1.
Find the coordinates of the point which divides the line segment joining the points (2,1) and (3,1) internally in the ratio 3:2.
Solution:
Let P(x, y) be the point which divides the line segment joining A(2,1) and B(3,1) internally in the ratio 2 : 1
Here,
(x1, y1) = (2,1)
(x2, y2) = (3,1)
m : n = 2 : 1
Using the section formula,
P(x, y) = (mx1+nx2/m + n , my1+my2/m + n)
(2×1 + 1×3/2 + 1, 2×1 + 1×1/2 + 1)
(2 + 3/3, 2 + 1/3)
(5/3, 3/3)
(5/3,1)
x = 5/3, y = 1

Example 2.
Find the coordinates of the points of trisection of the line segment joining the point (3,2) and the origin.
Solution:
Let P and Q be the points at trisection of the line segment joining A(3,2) and B(0,0). P provides AB in the ratio of 1:2
Therefore the coordinates of the point P are.
(1×0 + 2×3/1 + 2, 1×0 + 2×2/1 + 2)
(0+6/3,4/3
(2,4/3)
Q divides AB in the ratio 2: 1 therefore the coordinates of point Q are
(2×0 + 1×3/2+1, 2×0 + 1×2/2+1)
(3/3, 2/3)
(1,2/3)
Thus the required points are (3,2) and (1,4/3)

Example 3.
Given a triangle ABC in which A = (4,-4) B(0,5) and C(5,10). A point P lies on BC such that BP : PC = 3:2. Find the length of line segment AP.
Solution:
Given that
BP:PC = 3:2
Using the section formula the coordinates of a point P are
(3×5+2×0/3+2, 3×10+2×5/3+2)
(15/2,40/5)
(3,8)

Example 4.
The 4 vertices of a parallelogram are A(-4, 3), B(3, -1), C(p, q), and D(-1, 9). Find the value of p and q.
Solution:
Given vertices of a parallelogram are:
A(-4, 3), B(3, -1), C(p, q) and D(-1, 3)
We know that diagonals of a parallelogram bisect each other.
Let O be the point at which diagonals intersect.
Coordinates of mid-points of both AC and BD will be the same.
Therefore,
Using midpoint section formula,
(x1+x2/2 , y1+y2/2)
-4 + p/2 = -1 + 3/2
-4 + P = 2
P = 4 + 2
P = 6
Similarly
3 + q/2 = 3 – 1/ 2
3 + q = 2
q = 2-3
q = -1

Example 5.
Find the coordinates of the point which divides the line segment joining the points (4,2) and (5,1) internally in the ratio 3:2.
Solution:
Let P(x, y) be the point which divides the line segment joining A(4,2) and B(5,1) internally in the ratio 2 : 1
Here,
(x1, y1) = (4,1)
(x2, y2) = (5,1)
m : n = 2 : 1
Using the section formula,
P(x, y) = (mx1+nx2/m + n , my1+my2/m + n)
(2×4 + 1×5/2 + 1, 2×2 + 1×1/2 + 1)
(8 + 5/3, 4 + 1/3)
(13/3, 5/3)
(14/3,5/3)
x = 14/3, y = 5/3

FAQs on Section Formula

1. What are the Applications of Section Formula?

Section formula is used in various places in mathematics. In mathematics, we can use the section formula to find the centroid, incenters, and excenters of a triangle, etc. The section formula is also widely used to find the midpoint of a line segment.

2. What is the Section Formula for Internal Division?

If we have a line segment AB that is divided by a point P(x, y) internally in a ratio such that AP: PB = m: n,
Then the section formula for internal division is:
P(x, y) = (mx2+nx1/m+n,my2+ny1/m+n)
where,
x and y are the coordinates of point P
(x1, y1) are the coordinates of point A
(x2, y2) are the coordinates of the point B
m and n are the ratio values in which P divides the line internally

3. How is the section formula derived?

Section formula can be derived by constructing two right triangles and by using AA similarity. To find the ratio of the length of the sides of the triangle solve for x, and y, we can find the coordinates of the point that is dividing the line segment.

Midpoint Theorem on Right Angled Triangle

Midpoint Theorem on Right Angled Triangle – Statement & Proof | How do you find the Midpoint of a Right Angled Triangle?

This article aids the children to gain more knowledge about Midpoint Theorem on Right-angled Triangle. Usually, we know a triangle is cited as a polygon that has three sides of three line segments. In other words, a triangle is just a closed figure where the sum of its angles is equal to 180 degrees. Every shape of a triangle is classified based on the angle made by the two adjacent sides of a particular triangle.

The right-angle triangle is a geometrical shape, it plays a prominent role, and is considered as a basis in trigonometry. On this page, we will be proving that in a right-angled triangle the median drawn to the hypotenuse is half the hypotenuse in length. Let us discuss more on right-angle triangles and prove the statement of the midpoint theorem on the right-angled triangles with a few examples.

Right-angled Triangle – Definition

A right-angled triangle is defined as if one of the angles of a triangle is a 90 degrees right angle, then the triangle is called a right-angled triangle or simply can call a right triangle. In a triangle, the relation between the various sides can be easily realized with the help of a Pythagoras rule. These right-angles triangles are of two types namely, isosceles right-angled triangle and a scalene right-angled triangle.

The side opposite to the right angle is called hypotenuse and it is the largest side of the right-angle triangle. In the following image, triangle ABC is a right triangle, with three sides namely, the base, the altitude, and the hypotenuse.

Right-angled Triangle

 

 

 

 

 

In the above right-angled triangle, AB is the altitude, BC is the base, and the AC is the hypotenuse. AC is the largest side and it is opposite to the right angle within a triangle.

Midpoint of Right Angled Triangle Formula

The formula is represented as the square of the hypotenuse is equal to the sum of the square of the base and the square of the altitude.

Formula: (Hypotenuse)² = (Base)² + (Altitude)².

Midpoint Theorem on Right-Angled Triangle Statement & Proof

Prove that in a right-angled triangle the length of the median drawn to the hypotenuse is half the hypotenuse in length.

Right-angled Triangle with Median

Given:

In ∆ABC, ∠B = 90°, and BD is the median drawn to hypotenuse AB.

To Prove:

We have to prove that in a right-angled triangle ABC,

BD = ½ AC.

Construction:

From the above diagram ∆ABC, construct a right-angled triangle to prove the theorem statement.

Draw a line E and DE ∥ BC such that DE cuts AB at E.

Constructed right-angled triangle

Proof:

In ∆ABC, AD = ½ AC ( D is the midpoint of AC )

In ∆ABC,

(i) D is the midpoint of AC (by given)

(ii) DE ∥ BC (by construction)

Therefore, E is the midpoint of AB (by the converse of the midpoint theorem) —- (1)

ED ⊥ AB (since ED ∥ BC and BC ⊥ AB) —- (2)

In ∆AED and ∆BED,

(i) AE = EB (from (1))

(ii) ED = ED (common side)

(iii) ∠AED = ∠BED = 90° (from (2))

Therefore, ∆AED ≅ ∆BED (by SAS congruency theorem)

AD = BD (by c.p.c.t) —- (3)

Therefore, BD = ½ AC (using (3) in AD = ½ AC).

Hence, the theorem is proved.

Do Check:

Examples on Midpoint of Hypotenuse of Right-angled Triangles

Example 1:
In ∆PQR, QS is the median of ∆PQR. If the hypotenuse of PR= 12cm. Find the length of the median QS.

Example1 for Right-angled Triangle
Solution:
We know in a right-angled triangle, the length of the median of the hypotenuse is half the length of the hypotenuse.
Given PR= 12cm
Now, have to find the length of the median QS.
The formula used for a right-angled triangle is QS= ½PR.
QS= ½×12
⇒ QS= 6cm.
Therefore, the length of the median QS of ∆PQR is 6cm.

Example 2:
In ∆ABC, BD is the median of ∆ABC. If the length of the median BD is 14cm. Find the length of the hypotenuse AC.
Example2 for Right-angled Triangle
Solution: 
The statement used for a right-angled triangle is the median drawn to the hypotenuse is half the hypotenuse in length.
Given the length of the median BD= 14cm.
Now, find the length of the hypotenuse BD.
The formula we use is BD = ½AC.
14= ½ × AC
⇒ 14 × 2 = AC
⇒ AC= 28cm.
Therefore, the length of the hypotenuse AC of ∆ABC is 28cm.

FAQ’s on Midpoint of Right-Angled Triangle

1. What is the midpoint of the hypotenuse of a right triangle called?
The midpoint of the hypotenuse of a right-angled triangle or a right triangle is known as the circumcenter.

2. Does the median of a triangle form a right angle?
The median of a triangle does not always form a right angle to the side on which it is falling. In case, if we have an equilateral triangle or an isosceles triangle one median falls on the non-equal side of an isosceles triangle.

3. What is the midpoint of a right triangle?

The midpoint of the hypotenuse of a right triangle is the circumcenter of the triangle. Let assume any of the three points on the circle like A(a,0), B(b,0), and C(b,c) thus, the midpoint of the hypotenuse is equal to the center of the circle.

4. Why is the median half the hypotenuse?
Normally, the median of a triangle is drawn from one vertex to the midpoint of the opposite side of the vertex. In a right-angled triangle, the median to the hypotenuse has its property, i.e., the length of the hypotenuse is equal to half the length of the hypotenuse.