R S AGGARWAL AND V AGGARWAL Solutions for Class 9 Maths Chapter 4 - Linear Equations in Two Variables

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Chapter 4 - Linear Equations in Two Variables Exercise Ex. 4A

Question 1

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

3x + 5y = 7.5 

Solution 1

We have,

3x + 5y = 7.5

3x + 5y - 7.5 = 0

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

6x + 10y - 15 = 0

On comparing this equation with ax + by + c = 0, we obtain

a = 6, b = 10 and c = -15 

Question 2

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

On comparing this equation with ax + by + c = 0, we obtain

a = 10, b = -1 and c = 30 

Question 3

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

3y - 2x = 6

Solution 3

We have,

3y - 2x = 6

-2x + 3y - 6 = 0

On comparing this equation with ax + by + c = 0, we obtain

a = -2, b = 3 and c = -6 

Question 4

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

4x = 5y

Solution 4

We have,

4x = 5y

4x - 5y = 0

On comparing this equation with ax + by + c = 0, we obtain

a = 4, b = -5 and c = 0 

Question 5

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

 6x - 5y = 30

 6x - 5y - 30 = 0

On comparing this equation with ax + by + c = 0, we obtain

a = 6, b = -5 and c = -30 

Question 6

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

On comparing this equation with ax + by + c = 0, we obtain

a = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables, b = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables and c = -5 

Question 7

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

x = 6

Solution 7

We have,

x = 6

x - 6 = 0

1x + 0y - 6 = 0

x + 0y - 6 = 0

On comparing this equation with ax + by + c = 0, we obtain

a = 1, b = 0 and c = -6 

Question 8

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

3x - y = x - 1

Solution 8

We have,

3x - y = x - 1

3x - x - y + 1 = 0

2x - y + 1 = 0

On comparing this equation with ax + by + c = 0, we obtain

a = 2, b = -1 and c = 1 

Question 9

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

2x + 9 = 0

Solution 9

We have,

2x + 9 = 0

2x + 0y + 9 = 0

On comparing this equation with ax + by + c = 0, we obtain

a = 2, b = 0 and c = 9 

Question 10

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

4y = 7

Solution 10

We have,

4y = 7

0x + 4y - 7 = 0 

On comparing this equation with ax + by + c = 0, we obtain

a = 0, b = 4 and c = -7 

Question 11

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

x + y = 4

Solution 11

We have,

x + y = 4

x + y - 4 = 0 

On comparing this equation with ax + by + c = 0, we obtain

a = 1, b = 1 and c = -4 

Question 12

Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Solution 12

We have,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

3x - 8y - 1 = 0 

On comparing this equation with ax + by + c = 0, we obtain

a = 3, b = -8 and c = -1 

Question 13

Check which of the following are the solutions of the equation 5x - 4y = 20.

(4, 0)

Solution 13

Given equation is 5x - 4y = 20

Substituting x = 4 and y = 0 in L.H.S. of given equation, we get

L.H.S. = 5x - 4y

= 5(4) - 4(0)

= 20 - 0

= 20

= R.H.S.

Hence, (4, 0) is the solution of the given equation.

Question 14

Check which of the following are the solutions of the equation 5x - 4y = 20.

(0, 5)

Solution 14

Given equation is 5x - 4y = 20

Substituting x = 0 and y = 5 in L.H.S. of given equation, we get

L.H.S. = 5x - 4y

= 5(0) - 4(5)

= 0 - 20

= -20

R.H.S.

Hence, (0, 5) is not the solution of the given equation. 

Question 15

Check which of the following are the solutions of the equation 5x - 4y = 20.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Solution 15

Given equation is 5x - 4y = 20

Substituting x = -2 and y = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables in L.H.S. of given equation, we get

L.H.S. = 5x - 4y

= 5(-2) - 4R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

= -10 - 10

= -20

R.H.S.

Hence, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables is not the solution of the given equation. 

Question 16

Check which of the following are the solutions of the equation 5x - 4y = 20.

(0, -5)

Solution 16

Given equation is 5x - 4y = 20

Substituting x = 0 and y = -5 in L.H.S. of given equation, we get

L.H.S. = 5x - 4y

= 5(0) - 4(-5)

= 0 + 20

= 20

= R.H.S.

Hence, (0, -5) is the solution of the given equation. 

Question 17

Check which of the following are the solutions of the equation 5x - 4y = 20.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Solution 17

Given equation is 5x - 4y = 20

Substituting x = 2 and y = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables in L.H.S. of given equation, we get

L.H.S. = 5x - 4y

= 5(2) - 4R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

= 10 + 10

= 20

= R.H.S.

Hence, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables is the solution of the given equation. 

Question 18

Find five different solutions of each of the following equations:

2x - 3y = 6

Solution 18

Given equation is 2x - 3y = 6

Substituting x = 0 in the given equation, we get

2(0) - 3y = 6

0 - 3y = 6

3y = -6

y = -2

So, (0, -2) is the solution of the given equation.

 

Substituting y = 0 in the given equation, we get

2x - 3(0) = 6

2x - 0 = 6

2x = 6

x = 3

So, (3, 0) is the solution of the given equation.

 

Substituting x = 6 in the given equation, we get

2(6) - 3y = 6

12 - 3y = 6

3y = 6

y = 2

So, (6, 2) is the solution of the given equation.

 

Substituting y = 4 in the given equation, we get

2x - 3(4) = 6

2x - 12 = 6

2x = 18

x = 9

So, (9, 4) is the solution of the given equation.

 

Substituting x = -3 in the given equation, we get

2(-3) - 3y = 6

-6 - 3y = 6

3y = -12

y = -4

So, (-3, -4) is the solution of the given equation.

Question 19

Find five different solutions of each of the following equations:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Solution 19

Given equation is R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Substituting x = 0 in (i), we get

4(0) + 3y = 30

3y = 30

y = 10

So, (0, 10) is the solution of the given equation.

 

Substituting x = 3 in (i), we get

4(3) + 3y = 30

12 + 3y = 30

3y = 18

y = 6

So, (3, 6) is the solution of the given equation.

 

Substituting x = -3 in (i), we get

4(-3) + 3y = 30

-12 + 3y = 30

3y = 42

y = 14

So, (-3, 14) is the solution of the given equation.

 

Substituting y = 2 in (i), we get

4x + 3(2) = 30

4x + 6 = 30

4x = 24

x = 6

So, (6, 2) is the solution of the given equation.

 

Substituting y = -2 in (i), we get

4x + 3(-2) = 30

4x - 6 = 30

4x = 36

x = 9

So, (9, -2) is the solution of the given equation.

Question 20

Find five different solutions of each of the following equations:

3y = 4x

Solution 20

Given equation is 3y = 4x

Substituting x = 3 in the given equation, we get

3y = 4(3)

3y = 12

y = 4

So, (3, 4) is the solution of the given equation.

 

Substituting x = -3 in the given equation, we get

3y = 4(-3)

3y = -12

y = -4

So, (-3, -4) is the solution of the given equation.

 

Substituting x = 9 in the given equation, we get

3y = 4(9)

3y = 36

y = 12

So, (9, 12) is the solution of the given equation.

 

Substituting y = 8 in the given equation, we get

3(8) = 4x

4x = 24

x = 6

So, (6, 8) is the solution of the given equation.

 

Substituting y = -8 in the given equation, we get

3(-8) = 4x

4x = -24

x = -6

So, (-6, -8) is the solution of the given equation.

Question 21

If x = 3 and y = 4 is a solution of the equation 5x - 3y = k, find the value of k.

Solution 21

Since x = 3 and y = 4 is a solution of the equation 5x - 3y = k, substituting x = 3 and y = 4 in equation 5x - 3y = k, we get

5(3) - 3(4) = k

15 - 12 = k

k = 3 

Question 22

If x = 3k + 2 and y = 2k - 1 is a solution of the equation 4x - 3y + 1 = 0, find the value of k.

Solution 22

Since x = 3k + 2 and y = 2k - 1 is a solution of the equation 4x - 3y + 1 = 0, substituting these values in equation, we get

4(3k + 2) - 3(2k - 1) + 1 = 0

12k + 8 - 6k + 3 + 1 = 0

6k + 12 = 0

6k = -12

k = -2 

Question 23

The cost of 5 pencils is equal to the cost of 2 ballpoints. Write a linear equation in two variables to represent this statement. (Take the cost of a pencil to be Rs. x and that of a ballpoint to be Rs. y).

Solution 23

Let the cost of one pencil be Rs. x and that of one ballpoint be Rs. y.

Then,

Cost of 5 pencils = Rs. 5x

Cost of 2 ballpoints = Rs. 2y

According to given statement, we have

5x = 2y

5x - 2y = 0

Chapter 4 - Linear Equations in Two Variables Exercise Ex. 4B

Question 1

Draw the graph of each of the following equation.

x = 4

Solution 1

x = 4 is a line parallel to the Y-axis, at a distance of 4 units from it, to its right.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 2

Draw the graph of each of the following equation.

x + 4 = 0

Solution 2

x + 4 = 0

x = -4, which is a line parallel to the Y-axis, at a distance of 4 units from it, to its left.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 3

Draw the graph of each of the following equation.

y = 3

Solution 3

y = 3 is a line parallel to the X-axis, at a distance of 3 units from it, above the X-axis.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 4

Draw the graph of each of the following equation.

y = -3

Solution 4

y = -3 is a line parallel to the X-axis, at a distance of 3 units from it, below the X-axis.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 5

Draw the graph of each of the following equation.

x = -2

Solution 5

x = -2 is a line parallel to the Y-axis, at a distance of 2 units from it, to its left.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 6

Draw the graph of each of the following equation.

x = 5

Solution 6

x = 5 is a line parallel to the Y-axis, at a distance of 5 units from it, to its right.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 7

Draw the graph of each of the following equation.

y + 5 = 0 

Solution 7

y + 5 = 0

y = -5, which is a line parallel to the X-axis, at a distance of 5 units from it, below the X-axis.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 8

Draw the graph of each of the following equation.

y = 4

Solution 8

y = 4 is a line parallel to the X-axis, at a distance of 4 units from it, above the X-axis.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 9

Draw the graphs of the lines x - y = 1 and 2x + y = 8. Shade the area formed by these two lines and the y-axis. Also, find this area.

Solution 9

Graph of the equation x - y = 1

y = x - 1

When x = 1, then y = 1 - 1 = 0 

When x = 2, then y = 2 - 1 = 1

Thus, we have the following table:

x

1

2

y

0

1

 

Now, plot the points A(1, 0) and B(2, 1) on a graph paper.

Join AB and extend it in both the directions.

Then, the line AB is the required graph of x - y = 1.

 

Graph of the equation 2x + y = 8

y = 8 - 2x

When x = 2, then y = 8 - 2(2) = 8 - 4 = 4 

When x = 3, then y = 8 - 2(3) = 8 - 6 = 2 

Thus, we have the following table:

x

2

3

y

4

2

 

Now, plot the points C(2, 4) and D(3, 2) on a graph paper.

Join CD and extend it in both the directions.

Then, the line CD is the required graph of 2x + y = 8.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

The two graph lines intersect at point D(3, 2).

The area enclosed by the lines and Y-axis is shown in the graph.

Draw DM perpendicular from D on Y-axis.

DM = x-coordinate of point D(3, 2) = 3

And, EF = 9

Area of shaded region = Area of ΔDEF

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Question 10

Draw the graph for each of the equations x + y = 6 and x - y = 2 on the same graph paper and find the coordinates of the point where the two straight lines intersect.

*Back answer incorrect.

Solution 10

Graph of the equation x + y = 6

y = 6 - x

When x = 2, then y = 6 - 2 = 4 

When x = 3, then y = 6 - 3 = 3

Thus, we have the following table:

x

2

3

y

4

3

 

Now, plot the points A(2, 4) and B(3, 3) on a graph paper.

Join AB and extend it in both the directions.

Then, the line AB is the required graph of x + y = 6.

 

Graph of the equation x - y = 2

y = x - 2

When x = 3, then y = 3 - 2 = 1 

When x = 4, then y = 4 - 2 = 2 

Thus, we have the following table:

x

3

4

y

1

2

 

Now, plot the points C(3, 1) and D(4, 2) on a graph paper.

Join CD and extend it in both the directions.

Then, the line CD is the required graph of x - y = 2.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

The two graph lines intersect at point D(4, 2).

Question 11

Two students A and B contributed Rs. 100 towards the Prime Minister's Relief Fund to help the earthquake victims. Write a linear equation to satisfy the above data and draw its graph.

Solution 11

Let the amount contributed by students A and B be Rs. x and Rs. y respectively.

Total contribution = 100

x + y = 100

y = 100 - x

When x = 25, then y = 100 - 25 = 75

When x = 50, then y = 100 - 50 = 50

Thus, we have the following table:

x

25

50

y

75

50

 

Now, plot the points A(25, 75) and B(50, 50) on a graph paper.

Join AB and extend it in both the directions.

Then, the line AB is the required graph of x + y = 100.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 12

Draw the graph of the equation y = 3x.

From your graph, find the value of y when x = 2.

Solution 12

y = 3x

When x = 1, then y = 3(1) = 3

When x = -1, then y = 3(-1) = -3

Thus, we have the following table:

x

1

-1

y

3

-3

 

Now, plot the points A(1, 3) and B(-1, -3) on a graph paper.

Join AB and extend it in both the directions.

Then, the line AB is the required graph of y = 3x.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Reading the graph

Given: x = 2. Take a point M on the X-axis such that OM = 2.

Draw MP parallel to the Y-axis, cutting the line AB at P.

Clearly, PM = 6

Thus, when x = 2, then y = 6.

Question 13

Draw the graph of the equation y = 3x. From your graph, find the value of y when x = -2.

Solution 13

The given equation is y = 3x.

Putting x = 1, y = 3 1 = 3

Putting x = 2, y = 3 2 = 6

Thus, we have the following table:

x

1

2

y

3

6

Plot points (1,3) and (2,6) on a graph paper and join them to get the required graph.

Take a point P on the left of y-axis such that the distance of point P from the y-axis is 2 units.

Draw PQ parallel to y-axis cutting the line y = 3x at Q. Draw QN parallel to x-axis meeting y-axis at N.

So, y = ON = -6.

Question 14

Draw the graph of the equation x + 2y - 3 = 0. From your graph, find the value of y when x = 5.

Solution 14

The given equation is,

x + 2y - 3 = 0

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variablesx = 3 - 2y

Putting y = 1,x = 3 - (2 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables1) = 1

Putting y = 0,x = 3 - (2 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables0) = 3

Thus, we have the following table:

x

1

3

y

1

0

Plot points (1,1) and (3,0) on a graph paper and join them to get the required graph.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Take a point Q on x-axis such that OQ = 5.

Draw QP parallel to y-axis meeting the line (x = 3 - 2y) at P.

Through P, draw PM parallel to x-axis cutting y-axis at M.

So, y = OM = -1.

Question 15

Draw the graph of the equation x + 2y - 3 = 0.

From your graph, find the value of y when x = -5

Solution 15

x + 2y - 3 = 0

2y = 3 - x

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

When x = -1, then R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

When x = 1, then R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Thus, we have the following table:

x

-1

1

y

2

1

 

Now, plot the points A(-1, 2) and B(1, 1) on a graph paper.

Join AB and extend it in both the directions.

Then, the line AB is the required graph of x + 2y - 3 = 0.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Reading the graph

Given: x = -5. Take a point M on the X-axis such that OM = -5.

Draw MP parallel to the Y-axis, cutting the line AB at P.

Clearly, PM = 4

Thus, when x = -5, then y = 4. 

Question 16

Draw the graph of the equation 2x - 3y = 5. From the graph, find (i) the value of y when x = 4, and (ii) the value of x when y = 3.

Solution 16

The given equation is, 2x - 3y = 5

Now, if x = 4, then

And, if x = -2, then

Thus, we have the following table:

x

4

-2

y

1

-3

Plot points (4,1) and (-2,-3) on a graph paper and join them to get the required graph.

(i) When x = 4, draw a line parallel to y-axis at a distance of 4 units from y-axis to its right cutting the line at Q and through Q draw a line parallel to x-axis cutting y-axis which is found to be at a distance of 1 units above x-axis.

Thus, y = 1 when x = 4.

(ii) When y = 3, draw a line parallel to x-axis at a distance of 3 units from x-axis and above it, cutting the line at point P. Through P, draw a line parallel to y-axis meeting x-axis at a point which is found be 7 units to the right of y axis.

Thus, when y = 3, x = 7.

Question 17

Draw the graph of the equation 2x + y = 6. Find the coordinates of the point, where the graph cuts the x-axis.

Solution 17

The given equation is 2x + y = 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables y = 6 - 2x

Now, if x = 1, then y = 6 - 2 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 1 = 4

And, if x = 2, then y = 6 - 2 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 2 = 2

Thus, we have the following table:

x

1

2

y

4

2

Plot points (1,4) and (2,2) on a graph paper and join them to get the required graph.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

We find that the line cuts the x-axis at a point P which is at a distance of 3 units to the right of y-axis.

So, the co-ordinates of P are (3,0).

Question 18

Draw the graph of the equation 3x + 2y = 6. Find the coordinates of the point, where the graph cuts the y-axis.

Solution 18

The given equation is 3x + 2y = 6

2y = 6 - 3x

Now, if x = 2, then

And, if x = 4, then

Thus, we have the following table:

x

2

4

y

0

-3

Plot points (2, 0) and (4,-3) on a graph paper and join them to get the required graph.

We find that the line 3x + 2y = 6 cuts the y-axis at a point P which is 3 units above the x-axis.

So, co-ordinates of P are (0,3).

Question 19

Draw the graphs of the equations 3x - 2y = 4 and x + y - 3 = 0. On the same graph paper, find the coordinates of the point where the two graph lines intersect.

Solution 19

Graph of the equation 3x - 2y = 4

2y = 3x - 4

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

When x = 2, then R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

When x = -2, then R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Thus, we have the following table:

x

2

-2

y

1

-5

 

Now, plot the points A(2, 1) and B(-2, -5) on a graph paper.

Join AB and extend it in both the directions.

Then, the line AB is the required graph of 3x - 2y = 4.

 

Graph of the equation x + y - 3 = 0

y = 3 - x

When x = 1, then y = 3 - 1 = 2 

When x = -1, then y = 3 - (-1) = 4

Thus, we have the following table:

x

1

-1

y

2

4

 

Now, plot the points C(1, 2) and D(-1, 4) on a graph paper.

Join CD and extend it in both the directions.

Then, the line CD is the required graph of x + y - 3 = 0.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

The two graph lines intersect at point A(2, 1). 

Question 20

Draw the graph of the line 4x + 3y = 24.

Write the coordinates of the points where this line intersects the x-axis and the y-axis.

Solution 20

4x + 3y = 24

3y = 24 - 4x

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

When x = 0, then R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

When x = 3, then R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Thus, we have the following table:

x

0

3

y

8

4

 

Now, plot the points A(0, 8) and B(3, 4) on a graph paper.

Join AB and extend it in both the directions.

Then, the line AB is the required graph of 4x + 3y = 24.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Reading the graph

The graph of line 4x + 3y = 24 intersects the X-axis at point C(6, 0) and the Y-axis at point A(0, 8). 

Question 21

Draw the graph of the line 4x + 3y = 24.

Use this graph to find the area of the triangle formed by the graph line and the coordinate axes.

Solution 21

4x + 3y = 24

3y = 24 - 4x

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

When x = 0, then R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

When x = 3, then R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Thus, we have the following table:

x

0

3

y

8

4

 

Now, plot the points A(0, 8) and B(3, 4) on a graph paper.

Join AB and extend it in both the directions.

Then, the line AB is the required graph of 4x + 3y = 24.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Reading the graph

Required area = Area of ΔAOC

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Question 22

Draw the graphs of the lines 2x + y = 6 and 2x - y + 2 = 0. Shade the region bounded by these two lines and the x-axis. Find the area of the shaded region.

Solution 22

Graph of the equation 2x + y = 6

y = 6 - 2x

When x = 1, then y = 6 - 2(1) = 6 - 2 = 4 

When x = 2, then y = 6 - 2(2) = 6 - 4 = 2

Thus, we have the following table:

x

1

2

y

4

2

 

Now, plot the points A(1, 4) and B(2, 2) on a graph paper.

Join AB and extend it in both the directions.

Then, the line AB is the required graph of 2x + y = 6.

 

Graph of the equation 2x - y + 2 = 0

y = 2x + 2

When x = -1, then y = 2(-1) + 2 = -2 + 2 = 0 

When x = 2, then y = 2(2) + 2 = 4 + 2 = 6

Thus, we have the following table:

x

-1

2

y

0

6

 

Now, plot the points C(-1, 0) and D(2, 6) on a graph paper.

Join CD and extend it in both the directions.

Then, the line CD is the required graph of 2x - y + 2 = 0.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

The two graph lines intersect at point A(1, 4).

The area enclosed by the lines and X-axis is shown in the graph.

Draw AM perpendicular from A on X-axis.

PM = y-coordinate of point A(1, 4) = 4

And, CP = 4

Area of shaded region = Area of ΔACP

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Chapter 4 - Linear Equations in Two Variables Exercise MCQ

Question 1

The equation of the x-axis is

(a) x = 0

(b) y = 0

(c) x = y

(d) x + y = 0

Solution 1

Correct option: (b)

The equation of the x-axis is y = 0.

Question 2

The graph of y + 2 = 0 is a line

(a) making an intercept -2 on the x-axis

(b) making an intercept -2 on the y-axis

(c) parallel to the x-axis at a distance of 2 units below the x-axis

(d) parallel to the y-axis at a distance of 2 units to the left of y-axis

Solution 2

Correct option: (c)

The graph of y + 2 = 0 is a line parallel to the x-axis at a distance of 2 units below the x-axis. 

Question 3

The graph of the linear equation 2x + 3y = 6 meets the y-axis at the point

(a) (2, 0)

(b) (3, 0)

(c) (0, 2)

(d) (0, 3)

Solution 3

Correct option: (c)

When a graph meets the y-axis, the x coordinate is zero.

Thus, substituting x = 0 in the given equation, we get

2(0) + 3y = 6

3y = 6

y = 2

Hence, the required point is (0, 2).

Question 4

The graph of the linear equation 2x + 5y = 10 meets the x-axis at the point

(a) (0, 2)

(b) (2, 0)

(c) (5, 0)

(d) (0, 5)

Solution 4

Correct option: (c)

When a graph meets the x-axis, the y coordinate is zero.

Thus, substituting y = 0 in the given equation, we get

2x + 5(0) = 10

2x = 10

x = 5

Hence, the required point is (5, 0). 

Question 5

The graph of the line x = 3 passes through the point

(a) (0,3)

(b) (2,3)

(c) (3,2)

(d) None of these

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 6

The graph of the line y = 3 passes though the point

(a) (3, 0)

(b) (3, 2)

(c) (2, 3)

(d) none of these

Solution 6

Correct option: (c)

Since, the y coordinate is 3, the graph of the line y = 3 passes through the point (2, 3).

Question 7

The graph of the line y = -3 does not pass through the point

(a) (2,-3)

(b) (3,-3)

(c) (0,-3)

(d) (-3,2)

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two VariablesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 8

The graph of the linear equation x-y=0 passes through the point

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 9

If each of (-2,2), (0,0) and (2,-2) is a solution of a linear equation in x and y, then the equation is

(a) x-y=0

(b) x+y=0

(c) -x+2y=0

(d) x - 2y=0

Solution 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 10

How many linear equations can be satisfied by x = 2 and y = 3?

(a) only one

(b) only two

(c) only three

(d) Infinitely many

Solution 10

Correct option: (d)

Infinitely many linear equations can be satisfied by x = 2 and y = 3. 

Question 11

A linear equation in two variable x and y is of the form ax+by+c=0, where

(a) a0, b0

(b) a0, b=0

(c) a=0, b0

(d) a= 0, c=0

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 12

The equation of the y-axis is

(a) x = 0

(b) y = 0

(c) x = y

(d) x + y = 0

Solution 12

Correct option: (a)

The equation of the y-axis is x = 0. 

Question 13

If (2, 0) is a solution of the linear equation 2x + 3y = k then the value of k is

(a) 6

(b) 5

(c) 2

(d) 4

Solution 13

Correct option: (d)

Since, (2, 0) is a solution of the linear equation 2x + 3y = k, substituting x = 2 and y = 0 in the given equation, we have

2(2) + 3(0) = k

4 + 0 = k

k = 4 

Question 14

Any point on x-axis is of the form:

(a) (x,y), where x 0 and y 0

(b) (0,y), where y 0

(c) (x,0), where x 0

(d) (y,y), where y 0

Solution 14

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 15

Any point on y-axis is of the form

(a) (x,0), where x 0

(b) (0,y), where y 0

(c) (x,x), where x 0

(d) None of these

Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 16

x = 5, y = 2 is a solution of the linear equation

(a) x + 2y = 7

(b) 5x + 2y = 7

(c) x + y = 7

(d) 5x + y = 7

Solution 16

Correct option: (c)

Substituting x = 5 and y = 2 in L.H.S. of equation x + y = 7, we get

L.H.S. = 5 + 2 = 7 = R.H.S.

Hence, x = 5 and y = 2 is a solution of the linear equation x + y = 7. 

Question 17

If the point (3, 4) lies on the graph of 3y = ax + 7 then the value of a is

(a) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

(b) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

(c) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

(d) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables

Solution 17

Correct option: (b)

Since the point (3, 4) lies on the graph of 3y = ax + 7, substituting x = 3 and y = 4 in the given equation, we get

3(4) = a(3) + 7

12 = 3a + 7

3a = 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 18

The point of the form (a,a), where a 0 lies on

(a) x-axis

(b) y-axis

(c) the line y = x

(d) the line x + y = 0

Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 19

The point of the form (a,-a), where a 0 lies on

(a) x-axis

(b) y-axis

(c) the line y-x=0

(d) the line x + y = 0

Solution 19

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 20

The linear equation 3x - 5y = 15 has

(a) a unique solution

(b) two solutions

(c) infinitely many solutions

(d) no solution

Solution 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Linear Equations In Two Variables 

Question 21

The equation 2x + 5y = 7 has a unique solution, if x and y are

(a) natural numbers

(b) rational numbers

(c) positive real numbers

(d) real numbers

Solution 21

Correct option: (a)

The equation 2x + 5y = 7 has a unique solution, if x and y are natural numbers.

If we take x = 1 and y = 1, the given equation is satisfied. 

Question 22

The graph of y = 5 is a line

(a) making an intercept 5 on the x-axis

(b) making an intercept 5 on the y-axis

(c) parallel to the x-axis at a distance of 5 units from the origin

(d) parallel to the y-axis at a distance of 5 units from the origin

Solution 22

Correct option: (c)

The graph of y = 5 is a line parallel to the x-axis at a distance of 5 units from the origin. 

Question 23

The graph of x = 4 is a line

(a) making an intercept 4 on the x-axis

(b) making an intercept 4 on the y-axis

(c) parallel to the x-axis at a distance of 4 units from the origin

(d) parallel to the y-axis at a distance of 4 units from the origin

Solution 23

Correct option: (d)

The graph of x = 4 is a line parallel to the y-axis at a distance of 4 units from the origin. 

Question 24

The graph of x + 3 = 0 is a line

(a) making an intercept -3 on the x-axis

(b) making an intercept -3 on the y-axis

(c) parallel to the y-axis at a distance of 3 units to the left of y-axis

(d) parallel to the x-axis at a distance of 3 units below the x-axis

Solution 24

Correct option: (c)

The graph of x + 3 = 0 is a line parallel to the y-axis at a distance of 3 units to the left of y-axis.