# RS Aggarwal Solutions Class 10 Ex 3C

Question 1: On a graph paper, draw a horizontal line X’OX and a vertical line YoY’ as the x-axis and the y-axis respectively.

5x – y = 7, x – y + 1 = 0

Solution:

Given equations are 5x – y = 7 and x – y + 1 = 0

Graph of 5x – y = 7:

5x – y = 7 $\Rightarrow y = 5x – 7$

Putting x = 0, we get y = -7

Putting x = 1, we get y = -2

Putting x = 2, we get y = 3

Hence, the table is,

 x 0 1 2 y -7 -2 3

Plot the points A(0, 7), B(1, -2) and C(2, 3) on the graph paper. Join AB and BC to get the graph line AC. Extend it both ways.

Thus, line AC is the graph of 5x – y = 7.

Graph of x – y + 1 = 0:

x – y + 1 = 0 $\Rightarrow y =x+ 1$

Putting x = 0, we get y = 1

Putting x = 1, we get y = 2

Putting x = 2, we get y = 3

Hence, the table is,

 x 0 1 2 y 1 2 3

Now, on the same graph paper as above plot the points P(0, 1) and Q(1, 2). The point C(2, 3) has already been plotted. Join PA.

Thus, line PA is the graph of 3x + 2y – 12 = 0. The two graph lines intersect at C(2, 3).

Hence, x = 2, y = 3 is the solution of the given system of equations.

Clearly, the vertices of $\Delta APC$ formed by these lines and the y-axis are A(0, 7), P(0, 1) and C(2, 3).

Consider the triangle $\Delta APC$ :

Height of the triangle = 2 units and base(AP) = 8 units

Area of the triangle $\Delta APC$:

$Area = \frac{1}{2} \times Base \times height = \frac{1}{2} \times 8 \times 2$ sq. units

Area of $\Delta APC$ = 8 sq. units

Question 2: On a graph paper, draw a horizontal line X’OX and a vertical line YoY’ as the x-axis and the y-axis respectively.

x – 2y = 2, 4x – 2y = 5

Solution:

Given equations are x – 2y = 2 and 4x – 2y = 5

Graph of x – 2y = 2:

x – 2y = 2 $\Rightarrow y = \frac{x – 2}{2}$

Putting x = 0, we get y = -1

Putting x = 1, we get y = -0.5

Putting x = 2, we get y = 0

Hence, the table is,

 x 0 1 2 y -1 -0.5 0

Plot the points A(0, -1), B(2, 0) and C(1, -0.5) on the graph paper. Join AC and AB to get the graph line AB. Extend it both ways.

Thus, line AB is the graph of x – 2y = 2.

Graph of 4x – 2y = 5:

4x – 2y = 5 $\Rightarrow y = \frac{4x – 5}{2}$

Putting x = 0, we get y = -2.5

Putting x = 1, we get y = -0.5

Putting x = 2, we get y = -1.5

Hence, the table is,

 x 0 1 2 y -2.5 -0.5 -1.5

Now, on the same graph paper as above plot the points P(0, -0.25) and Q(2, -1.5). The point C(1, -0.5) has already been plotted. Join PC and CQ to get PQ.

Thus, line PA is the graph of 4x – 2y = 5. The two graph lines intersect at (1, -0.5).

The given system of equations is consistent.

Question 3: On a graph paper, draw a horizontal line X’OX and a vertical line YOY’ as the x-axis and the y-axis respectively.

2x + 3y = 4, 4x + 6y = 12

Solution:

Given equations are 2x + 3y = 4 and 4x + 6y = 12

Graph of 2x + 3y = 4:

2x + 3y = 4 $\Rightarrow y = \frac{-2x + 4}{3}$

Putting x = 2, we get y = 2

Putting x = -1, we get y = 2

Putting x = -4, we get y = 4

Hence, the table is,

 x 2 -1 -4 y 0 2 4

Plot the points A(2, 0), B(-1, 2) and C(-4, 4) on the graph paper. Join AB and BC to get the graph line AC. Extend it both ways.

Thus, line AC is the graph of 2x + 3y = 4.

Graph of 4x + 6y = 12:

4x + 6y = 12 $\Rightarrow y = \frac{-4x + 12}{6}$

Putting x = 3, we get y = 0

Putting x = 0, we get y = 2

Putting x = 6, we get y = -2

Hence, the table is,

 x 3 0 6 y 0 2 -2

Now, on the same graph paper as above plot the points P(3, 0) and Q(0, 2) and R(6, -2). Join PQ and PR to get QR.

Thus, line QR is the graph of 4x + 6y = 12.

It is not clear from the graph that two graph lines are parallel and do not intersect when produced.

Hence, the given system of equation is consistent.

Question 4: On a graph paper, draw a horizontal line X’OX and a vertical line YOY’ as the x-axis and the y-axis respectively.

2y – x = 9, 4y – 2x = 20

Solution:

Given equations are 2y – x = 9 and 4y – 2x = 20

Graph of 2y – x = 9:

2y – x = 9 $\Rightarrow y = \frac{x + 9}{2}$

Putting x = 1, we get y = 5

Putting x = -1, we get y = 4

Putting x = -3, we get y = 3

Hence, the table is,

 x 1 -1 -3 y 5 4 3

Plot the points A(1, 5), B(-1, 4) and C(-3, 3) on the graph paper. Join AB and BC to get the graph line AC. Extend it both ways.

Thus, line AC is the graph of 2y – x = 9.

Graph of 4y – 2x = 20:

4y – 2x = 20 $\Rightarrow y = \frac{2x + 20}{4}$

Putting x = 0, we get y = 5

Putting x = 2, we get y = 6

Putting x = -2, we get y = 4

Hence, the table is,

 x 0 2 -2 y 5 6 4

Now, on the same graph paper as above plot the points P(0, 5) and Q(2, 6) and R(-2, 4). Join PQ and PR to get QR.

Thus, line QR is the graph of 4y – 2x = 20.

It is not clear from the graph that two graph lines are parallel and do not intersect when produced.

Hence, the given system of equation is consistent.

Question 5: On a graph paper, draw a horizontal line X’OX and a vertical line YOY’ as the x-axis and the y-axis respectively.

3x – y = 5, 6x – 2y = 10

Solution:

Given equations are 3x – y = 5 and 6x – 2y = 10

Graph of 3x – y = 5:

3x – y = 5 $\Rightarrow y = 3x – 5$

Putting x = 1, we get y = -2

Putting x = 0, we get y = -5

Putting x = 2, we get y = 1

Hence, the table is:

 x 1 0 2 y -2 -5 1

Plot the points A(1, -2), B(0, -5) and C(2, 1) on the graph paper. Join AB and AC to get the graph line BC. Extend it both ways.

Thus, line BC is the graph of 3x – y = 5.

Graph of 6x – 2y = 10:

6x – 2y = 10 $\Rightarrow y = \frac{6x – 10}{2}$

Putting x = 0, we get y = -5

Putting x = 1, we get y = -2

Putting x = 2, we get y = 1

Hence, the table is:

 x 0 1 2 y -5 -2 1

These points are same as obtained above:

From the graph, it is clear that these two lines coincide. Both equations represent same graph. Hence, these lines have infinitely many solutions.

Question 6: On a graph paper, draw a horizontal line X’OX and a vertical line YOY’ as the x-axis and the y-axis respectively.

2x + y = 6, 6x + 3y = 18

Solution:

Given equations are 2x + y = 6 and 6x + 3y = 18

Graph of 2x + y = 6:

2x + y = 6 $\Rightarrow y = -2x + 6$

Putting x = 3, we get y = 0

Putting x = 1, we get y = 4

Putting x = 2, we get y = 2

Hence, the table is,

 x 3 1 2 y 0 4 2

Plot the points A(3, 0), B(1, 4) and C(2, 2) on the graph paper. Join AC and BC to get the graph line of equation 2x + y = 6.

Graph of 6x + 3y = 18:

6x + 3y = 18 $\Rightarrow y = \frac{-6x + 18}{3}$

Putting x = 0, we get y = -5

Putting x = 1, we get y = -2

Putting x = 2, we get y = 1

Hence, the table is:

 x 3 1 2 y 0 4 2

These points are same as obtained above:

Thus, we find that the two line graphs coincide. Hence, the given system of equations has infinitely many solutions.

Solve for x and y

Question 7: x + y = 8, 2x – 3y = 1

Solution:

The given equations are.

x + y = 8 …..(1)

2x – 3y = 1 …..(2)

Multiplying (1) by 3 and (2) by 1, we get,

3x + 3y = 24 …..(3)

2x – 3y = 1 …..(4)

Adding (3) and (4), we get

$5x \, = \, 25 \Rightarrow x \, = \, \frac{25}{5} \Rightarrow x \, = \, 5$

Substituting x = 5 in (1), we get

5 + y = 8 $\Rightarrow$ y = 8 – 5 = 3

∴ x = 5 and y = 3

Question 8: x + y = 3, 4x – 3y = 26

Solution:

The given equations are.

x + y = 3 …..(1)

4x – 3y = 26 …..(2)

Multiplying (1) by 3 and (2) by 1, we get,

3x + 3y = 9 …..(3)

4x – 3y = 26 …..(4)

Adding (3) and (4), we get

7x = 35 $\Rightarrow$ x = 5

Substituting x = 5 in (1), we get

5 + y = 3 $\Rightarrow$ y = 3 – 5 = -2

∴ x = 5 and y = -2

Question 9: x- y = 3, 3x – 2y

Solution:

The given equations are.

x- y = 3 …..(1)

3x – 2y = 10 …..(2)

Multiplying (1) by 2 and (2) by 1, we get,

x – y = 3 …..(3)

3x – 2y = 10 …..(4)

Subtracting (3) from (4), we get

x = 4

Substituting x = 4 in (1), we get

4 – y = 3 $\Rightarrow$ y = 4 – 3 = 1

∴ x = 4 and y = 3

Question 10: 2x + 3y = 0, 3x + 4y = 5

Solution:

The given equations are.

2x + 3y = 0 …..(1)

3x + 4y = 5 …..(2)

Multiplying (1) by 4 and (2) by 3, we get,

8x + 12y = 0 …..(3)

9x + 12y = 15 …..(4)

Subtracting (3) from (4), we get

x = 15

Substituting x = 15 in (1), we get

(2)(15) + 3y = 0 $\Rightarrow$ 3y = 0 – 30

3y = -30 or y = -10

∴ x = 15 and y = -10

Question 11: 2x – 3y = 13, 7x – 2y = 20

Solution:

The given equations are.

2x – 3y = 13 …..(1)

7x – 2y = 20 …..(2)

Multiplying (1) by 2 and (2) by 3, we get,

4x – 6y = 26 …..(3)

21x – 6y = 60 …..(4)

Subtracting (3) from (4), we get

17x = 34 $\Rightarrow$ x = 2

Substituting x = 2 in (1), we get

(2)(2) – 3y = 13 $\Rightarrow$ 4 – 3y = 13

-3y = 13 – 4 or -3y = 9 or y = -3

∴ x = 2 and y = -3

#### Practise This Question

If white ray falls on the system in the give way then how many colours will we see after all dispersion has happened?