Class 9 RD Sharma Solutions Chapter 13 Linear Equation in Two Variable Exercise 13.3 | Set 2 (original) (raw)

Last Updated : 30 Apr, 2021

Question 11: Draw the graph of the equation 2x + 3y = 12. From the graph, find the coordinates of the point:

(i) whose y-coordinates is 3.

(ii) whose x-coordinate is −3.

Solution:

Given:

2x + 3y = 12

We get,

y=\frac{12-2x}{3}

Substituting x = 0 in y = \frac{12-2x}{3}we get,

y=\frac{12-2\times0}{3}

y = 4

Substituting x = 6 in y= \frac{12-2x}{3}, we get

y=\frac{12-2\times6}{3}

y = 4

Hence, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	6
y	4	0

By plotting the given equation on the graph, we get the point B (0, 4) and C (6,0).

(i) Co-ordinates of the point whose y axis is 3 are A \left(\frac{3}{2},3\right)

(ii) Co-ordinates of the point whose x -coordinate is –3 are D (-3, 6)

Question 12: Draw the graph of each of the equations given below. Also, find the coordinates of the points where the graph cuts the coordinate axes:

(i) 6x − 3y = 12

(ii) −x + 4y = 8

(iii) 2x + y = 6

(iv) 3x + 2y + 6 = 0

Solution:

(i) Given:

6x - 3y = 12

We get,

y=\frac{6x-12}{3}

Now, substituting x = 0 in y=\frac{6x-12}{3}, we get

y = -4

Substituting x = 2 in y=\frac{6x-12}{3}, we get

y = 0

Hence, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	2
y	-4	0

Co-ordinates of the points where graph cuts the co-ordinate axes are at y = -4 axis and x = 2

at x axis.

(ii) Given:

-x + 4y = 8

We get,

y=\frac{8+x}{4}

Now, substituting x = 0 in y=\frac{8+x}{4}, we get

y = 2

Substituting x = -8 in y=\frac{8+x}{4}, we get

y = 0

Hence, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	-8
y	2	0

Co-ordinates of the points where graph cuts the co-ordinate axes are at y = 2 axis and x = -8

at x axis.

(iii) Given:

2x + y = 6

We get,

y = 6 - 2x

Now, substituting x= 0 in y = 6 - 2x w e get

y = 6

Substituting x = 3 in y = 6 - 2x, we get

y = 0

Hence, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	3
y	6	0

Co-ordinates of the points where graph cuts the co-ordinate axes are y = 6 at y axis and x = 3

at x axis.

(iv) Given:

3x + 2y + 6 = 0

We get,

y = \frac{-(6+3x)}{2}

Now, substituting x = -2 in y = \frac{-(6+3x)}{2}, we get

y = -3

Substituting x = -2 in y = \frac{-(6+3x)}{2}, we get

y = 0

Hence, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	-2
y	-3	0

Co-ordinates of the points where graph cuts the co-ordinate axes are y = -3 at y axis and x = -2

at x axis.

Question 13: Draw the graph of the equation 2x + y = 6. Shade the region bounded by the graph and the coordinate axes. Also, find the area of the shaded region.

Solution:

Given:

2x + y = 6

We get,

y = 6 - 2 x

Now, substituting x = 0 in y = 6 - 2x ,we get

y = 6

Substituting x = 3 in y = 6 - 2x ,we get

Hence, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	3
y	6	0

The region bounded by the graph is ABC which forms a triangle.

AC at y axis is the base of triangle having AC = 6 units on y axis.

BC at x axis is the height of triangle having BC = 3 units on x axis.

Therefore,

Area of triangle ABC, say A is given by

A=\frac{1}{2}(Base\times Height)\\ A=\frac{1}{2}(AC\times BC)\\ A=\frac{1}{2}(6\times3)

A = 9 sq. units

Question 14: Draw the graph of the equation \frac{x}{3}+\frac{y}{4}=1. Also, find the area of the triangle formed by the line and the coordinates axes.

Solution:

Given:

\frac{x}{3}+\frac{y}{4}=1

4x + 3y = 12

We get,

y=\frac{12-4x}{3}

Now, substituting x = 0 in y=\frac{12-4x}{3}, we get

y = 4

Substituting x = 3 in y=\frac{12-4x}{3}, we get

y = 0

Hence, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	3
y	4	0

The region bounded by the graph is ABC which form a triangle.

AC at y axis is the base of triangle having AC = 4 units on y axis.

BC at x axis is the height of triangle having BC = 3 units on x axis.

Therefore,

Area of triangle ABC, say A is given by

A=\frac{1}{2}(Base\times Height)\\ A=\frac{1}{2}(AC\times BC)\\ A=\frac{1}{2}(4\times3)

A = 6 sq. units

Question 15: Draw the graph of y = | x |.

Solution:

We are given,

y = |x|

Substituting, x = 1 we get

y = 1

Substituting, x = -1 we get

y = 1

Substituting,x = 2 we get

y = 2

Substituting, x = -2 we get

y = 2

For every value of x, whether positive or negative, we get y as a positive number.

Question 16: Draw the graph of y = | x | + 2.

Solution:

Given:

y = |x| + 2

Substituting, x = 0 we get

y = 2

Substituting, x = 1 we get

y = 3

Substituting, x = -1 we get

y = 3

Substituting, x = 2 we get

y = 4

Substituting, x = -2 we get

y = 4

For every value of x, whether positive or negative, we get y as a positive number and the minimum value of y is equal to 2 units.

Question 17: Draw the graphs of the following linear equations on the same graph paper:

2x + 3y = 12, x − y = 1

Find the coordinates of the vertices of the triangle formed by the two straight lines and the area bounded by these lines and x-axis.

Solution:

Given:

2x + 3y = 12

We get,

y=\frac{12-2x}{3}

Now, substituting x = 0 in y=\frac{12-2x}{3} , we get

y = 4

Substituting x = 6 in y=\frac{12-2x}{3} , we get

y = 0

Therefore, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	6
y	4	0

Plotting A(0,4) and E(6,0) on the graph and by joining the points , we obtain the graph of equation .

Given:

x - y = 1

We get,

y = x − 1

Now, substituting x = 0 in y = x − 1,we get

y = −1

Substituting x = -1 in y = x − 1,we get

y = −2

Therefore, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	-1
y	-1	-2

Plotting D(0,1) and E(-1,0) on the graph and by joining the points , we obtain the graph of equation .

By the intersection of lines formed by 2x + 3y = 12 and x - y = 1 on the graph, triangle ABC is formed on y axis.

Therefore,

AC at y axis is the base of triangle ABC having AC = 5 units on y axis.

Draw FE perpendicular from F on y axis.

FE parallel to x axis is the height of triangle ABC having FE = 3 units on x axis.

Therefore,

Area of triangle ABC, say A is given by

A=\frac{1}{2}(Base\times Height)\\ A=\frac{1}{2}(AC\times BC)\\ A=\frac{1}{2}(5\times3)\\ =\frac{15}{2}sq.\ units

Question 18: Draw the graphs of the linear equations 4x − 3y + 4 = 0 and 4x + 3y −20 = 0. Find the area bounded by these lines and x-axis.

Solution:

Given:

4x - 3y + 4 = 0

We get,

y=\frac{4x+4}{3}

Now, substituting x = 0 in y=\frac{4x+4}{3}, we get

y=\frac{4}{3}

Substituting x = -1 in y=\frac{4x+4}{3}, we get

y = 0

Hence, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	-1
y	\frac{4}{3}	0

Plotting E(0, \frac{4}{3}) and A(-1,0) on the graph and by joining the points, we obtain the graph of equation.

We are given,

4x + 3y - 20 = 0

We get,

y=\frac{20-4x}{3}

Now, substituting x = 0 in y=\frac{20-4x}{3}, we get

y=\frac{20}{3}

Substituting x = 5 in y=\frac{20-4x}{3}, we get

y = 0

Therefore, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	5
y	\frac{20}{3}	0

Plotting D(0, \frac{20}{3}) and B(5,0) on the graph and by joining the points, we obtain the graph of equation.

By the intersection of lines formed by 4x - 3y + 4 = 0 and 4x + 3y - 20 = 0 on the graph, triangle ABC is formed on x axis.

Therefore,

AB at x axis is the base of triangle ABC having AB = 6 units on x axis.

Draw CF perpendicular from C on x axis.

CF parallel to y axis is the height of triangle ABC having CF = 4 units on y axis.

Therefore,

Area of triangle ABC, say A is given by

A=\frac{1}{2}(Base\times Height)\\ A=\frac{1}{2}(AC\times BC)\\ A=\frac{1}{2}(6\times4)\\ A=12sq.\ units

Question 19: The path of a train A is given by the equation 3x + 4y − 12 = 0 and the path of another train B is given by the equation 6x + 8y − 48 = 0. Represent this situation graphically.

Solution:

Given: Path of train A,

3x + 4y - 12 = 0

We get,

y=\frac{12-3x}{4}

Now, substituting x = 0 in y=\frac{12-3x}{4} ,we get

y = 3

Substituting x = 4 in y=\frac{12-3x}{4} ,we get

y = 0

Therefore, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	4
y	3	0

Plotting A(4,0) and E(0,3) on the graph and by joining the points , we obtain the graph of equation .

We are given the path of train B,

6x + 8y - 48 = 0

We get,

y=\frac{48-6x}{8}

Now, substituting x = 0 in y=\frac{48-6x}{8},we get

y = 6

Substituting x = 8 in y=\frac{48-6x}{8},we get

y = 0

Therefore, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	8
y	6	0

Plotting C(0,6) and D(8,0) on the graph and by joining the points , we obtain the graph of equation.

Question 20: Ravish tells his daughter Aarushi, ''Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be''. If present ages of Aarushi and Ravish are x and y years respectively, represent this situation algebraically as well as graphically.

Solution:

We are given the present age of Ravish as y years and Aarushi as x years.

Age of Ravish seven years ago = y - 7

Age of Aarushi seven years ago = x - 7

It has already been said by Ravish that seven years ago he was seven times old then Aarushi was then

So,

y - 7 = 7(x - 7)

y - 7 = 7x - 49

7x - y = -7 + 49

7x - y - 42 = 0

Age of Ravish three years from now = y + 3

Age of Aarushi three years from now = x + 3

It has already been said by Ravish that three years from now he will be three times old then Aarushi will be then

So,

y + 3 = 3(x + 3)

y + 3 = 3x + 9

3x + 9 - y - 3 = 0

3x - y + 6 = 0

(1) and (2) are the algebraic representation of the given statement.

Given:

7x - y - 42 = 0

We get,

y = 7x - 42

Now, substituting x = 0 in y = 7x - 42 ,we get

y = -42

Substituting x = 6 in y = 7x - 42, we get

y = 0

Therefore, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	6
y	-42	0

Given:

3x - y + 6 = 0

We get,

y = 3x + 6

Now, substituting x = 0 in y = 3x + 6 ,we get

y = 6

Substituting x = -2 in y = 3x + 6 ,we get

y = 0

Therefore, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x	0	-2
y	6	0

The red -line represents the equation. 7x - y - 42 = 0

The blue-line represents the equation. 3x - y + 6 = 0

Question 21: Aarushi was driving a car with uniform speed of 60km/h. Draw distance-time graph. From the graph, find the distance traveled by Aarushi in

(i) 2\frac{1}{2} Hours

(ii) \frac{1}{2} Hour

Solution:

Aarushi is driving the car with the uniform speed of 60 km/h.

We represent time on X-axis and distance on Y-axis

Now, graphically

We are given that the car is travelling with a uniform speed 60 km/hr. This means car travels 60 km distance each hour. Thus the graph we get is of a straight line.

Also, we know when the car is at rest, the distance traveled is 0 km, speed is 0 km/hr and the time is also 0 hr.

Therefore, the given straight line will pass through O(0,0) and M(1,60).

Join the points O and M and extend the line in both directions.

Now, we draw a dotted line parallel to y-axis from x = \frac{1}{2} that meets the straight line graph at L from which we draw a line parallel to x-axis that crosses y-axis at 30. Thus, in \frac{1}{2}hr, distance traveled by the car is 30 km.

Now, we draw a dotted line parallel to y-axis from x = 2\frac{1}{2} that meets the straight line graph at N from which we draw a line parallel to x-axis that crosses y-axis at 150. Thus, in 2\frac{1}{2}hr, distance traveled by the car is 150 km.

(i) Distance = Speed × Time

Distance traveled in 2\frac{1}{2} hours is given by

Distance = 60\times2\frac{1}{2}

Distance = 60\times\frac{5}{2}

Distance = 150 km

(ii) Distance = Speed × Time

Distance traveled in \frac{1}{2} hours is given by

Distance = 60\times\frac{1}{2}

Distance = 30 km