Chapter 13 Linear Equations in Two Variables 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 – y = 0 and 3x – 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 − 1)
y = 15 + 8x − 8
y = 8x + 7
Where, Rs y is the total fare (x – 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.
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 by
x+y(7-3) = 27
⇒ x+4y = 27
⇒ x+y(7-3)=27
⇒ x+4y=27
Here, (7, -3) 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.
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,
10y + x = 10x + y +27
⇒ 10y- y+x-10x = 27
⇒ 9y-9x = 27
⇒ 9(y- x) = 27
⇒ y-x = 27/9
⇒ y-x=3
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 (−1, 3) on a graph paper and verify that the straight line passing through these points also passes through the point (1, 4).
Solutions
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 (–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
We are given co-ordinates (1, –1) and (–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 −2
(iii) y = x + 2
(iv) x + 2y = 6
Solution
We are given co-ordinates (–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.
Solution
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
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
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
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.
Solution
15. Draw the graph of y = | x |.
Solution
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 − 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 − 3y + 4 = 0 and 4x + 3y −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 − 12 = 0 and the path of another train B is given by the equation 6x + 8y − 48 = 0. Represent this situation graphically.
Solution
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
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) 2½ Hours
(ii) 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
