#### Chapter 13Â Linear Equations in Two VariablesÂ R.D. Sharma Solutions for Class 9th Exercise 13.3

**Exercise 13.3**

(i) x+y = 4

(ii) x-y = 0

(iii) -x+y = 6

(iv) y = 2x

(v) 3x+5y = 15

(vi) x/2 - y/3 = 2

(vii) (x-2)/3 = y-3

(viii) 2y = -x+1

**Solution**

2 . Give the equations of two lines passing through (3, 12). How many more such lines are there, and why?

**Solution**

We observe that x = 3 and y = 12 is the solution of the following equations

4x-y = 0 and 3x-y+3 = 0

So, we get the equations of two lines passing through (3, 12) are, 4x â€“ y = 0 and 3x â€“ y + 3 = 0.

We know that passing through the given point infinitely many lines can be drawn. So, there are infinitely many lines passing through (3,12)

3. A three-wheeler scooter charges Rs 15 for first kilometer and Rs 8 each for every subsequent kilometer. For a distance of x km, an amount of Rs y is paid. Write the linear equation representing the above information.

**Solution**

Total fare of Rs y for covering the distance of x km is given by

y = 15 + 8(x âˆ’ 1)

y = 15 + 8x âˆ’ 8

y = 8x + 7

Where, Rs y is the total fare (x â€“ 1) is taken as the cost of first kilometer is already given Rs 15 and 1 has to subtracted from the total distance travelled to deduct the cost of first kilometer.

4. A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Aarushi paid Rs 27 for a book kept for seven days. If fixed charges are Rs x and per day charges are Rs y. Write the linear equation representing the above information.

Â

**Solution**

Total charges of Rs 27 of which Rs x for first three days and Rs y per day for 4 more days is given byÂ

x+y(7-3) = 27

â‡’ x+4y = 27

â‡’ x+y(7-3)=27

â‡’ x+4y=27

Here, (7, -3)Â is taken as the charges for the first three days are already given at Rs x and we have to find the charges for the remaining four days as the book is kept for the total of 7 days.

5. A number is 27 more than the number obtained by reversing its digits. If its unit's and ten's digit are x and y respectively, write the linear equation representing the above statement.

**Solution**

The number given to us is in the form of yx,

where y represents the ten's place of the number and x represents the unit's place of the number.Â

Now,

Given number is 10y + x

Number obtained by reversing the digits of the number is 10x + y

It is given to us that the original number is 27 more than the number obtained by reversing its digits
So,Â

10y + x = 10x + y +27

â‡’ 10y- y+x-10x = 27Â

â‡’ 9y-9x = 27Â

â‡’ 9(y- x) = 27

â‡’ y-x = 27/9

â‡’ y-x=3Â

6. The sum of a two digit number and the number obtained by reversing the order of its digits is 121. If units and ten's digit of the number are x and y respectively, then write the linear equation representing the above statement.

**Solution**

7. Plot the points (3,5) and (âˆ’1, 3) on a graph paper and verify that the straight line passing through these points also passes through the point (1, 4).

**Solutions**

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By plotting the given points (3, 5) and (â€“1, 3) on a graph paper, we get the line BC.We have already plotted the point A (1, 4) on the given plane by the intersecting lines.

Therefore, it is proved that the straight line passing through (3, 5) and (â€“1, 3) also passes through A (1, 4).

8. From the choices given below, choose the equation whose graph is given in Fig. 13.13.

(i) y = x

(ii) x + y = 0

(iii) y = 2x

(iv) 2 + 3y = 7x

**Solution**Â

We are given co-ordinates (1, â€“1) and (â€“1, 1) as the solution of one of the following equations.

We will substitute the value of both co-ordinates in each of the equation and find the equation which satisfies the given co-ordinates.

(i) y= x + 2

(ii) y = x âˆ’2

(iii) y = x + 2

(iv) x + 2y = 6

**Solution**Â

We are given co-ordinates (â€“1, 3) and (2, 0) as the solution of one of the following equations.

We will substitute the value of both co-ordinates in each of the equation and find the equation which satisfies the given co-ordinates.
Â

**SolutionÂ**

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.

**Â SolutionÂ**

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 âˆ’ 3y = 12

(ii) -x + 4y = 8

(iii) 2x + y = 6

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

**Solution**Â

13. Draw the graph of the equation 2

*x*Â +Â*y*Â = 6. Shade the region bounded by the graph and the coordinate axes. Also, find the area of the shaded region.**Solution**Â

14. Draw the graph of the equation x/3 + y/4 = 1. Also, find the area of the triangle formed by the line and the coordinates axes.

**Â SolutionÂ**

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

**SolutionÂ**

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

**Solution**

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

2x + 3y = 12, x âˆ’ 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**

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

**Solution**

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

**Solution**Â

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**Â

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

(i) 2Â½ Hours

(ii) 1/2 Hour

**SolutionÂ**

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