Chapter 3 - Pair Of Linear Equations In Two Variables, RD Sharma Solutions - (Part-12) | RD Sharma Solutions for Class 10 Mathematics PDF Download

Page No 3.31

Q.37. Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are :
(i) y = x, y = 2x and y + x = 6
(ii) y = x, 3y = x, x + y = 8
Ans. (i) The given equations are
y = x   ......(i)
y = 2x  ......(ii)
y + x = 6  ......(iii)
The two points satisfying (i) can be listed in a table as,

The two points satisfying (ii) can be listed in a table as,
 
The two points satisfying (iii) can be listed in a table as,

Now, graph of equations (i), (ii) and (iii) can be drawn as,

It is seen that the coordinates of the vertices of the obtained triangle are A(0,0),B(2,4),C(3,3)
(ii) The given equations are
y = x    ......(i)
3y = x    ......(ii)
x + y = 8    ......(iii)
The two points satisfying (i) can be listed in a table as,

The two points satisfying (ii) can be listed in a table as, 

The two points satisfying (iii) can be listed in a table as,

Now, graph of equations (i), (ii) and (iii) can be drawn as,

It is seen that the coordinates of the obtained triangle are A(0,0), B(4,4), C(6,2)

Q.38. Graphically, solve the following pair of equations:
2x + y = 6
2x − y + 2 =0
Find the ratio of the areas of the two triangles formed by the lines representing these equations with the x-axis and the lines with the y-axis.
Ans. The given linear equations are:
2x + y = 6  .....(i)
2x − y + 2 = 0   .....(ii)
For (i), we have

For (ii), we have

Thus, we plot the graph for these two equations and mark the point where these two lines intersect.

From the graph we see that the two lines intersect at point E(1, 4).
Now, the area of triangle CEB is

The area of triangle AED is

So, the ratio of the areas of the two triangles will be

Thus, the required ratio is 4 : 1.

Q.39. Determine, graphically, the vertices of the triangle formed by the lines y = x, 3y = x , x + y = 8.
Ans. The given lines are:
y = x          .....(i)
3y = x        .....(ii)
x + y = 8    .....(iii)
From (i), we have

For (ii), we have

For (iii), we have

Thus, we plot the graph for these three equations and mark the point where these two lines intersect.

From the graph we find that the vertices of the triangle thus formed are H(4, 4), I(6, 2) and D(0, 0).

Q.40. Draw the graph of the equations  x = 3 , x = 5  and  2x − y − 4 = 0 . Also, find the area of the quadrilateral formed by the lines and the x-axis.
Ans. The given equations are
x = 3  .....(i)
x = 5  .....(ii)
2x − y − 4 = 0  .....(iii)
For (iii), we have

We plot the lines on the graph as follows:

The vertices of the quadrilateral thus formed are A(5, 0), B(5, 6), C(3, 2) and D(3, 0).
∴ Area of the quadrilateral DABC
= Area of a trapezium

= 8 square units
Thus, the area of the quadrilateral formed by the given lines and the x-axis is 8 square units.

Q.41. Draw the graph of the lines x = −2 and y = 3 . Write the vertices of the figure  formed by these lines , the x-axis  and the y-axis . Also , find the area of the figure.
Ans. The given lines are
x = − 2     .....(i)
y = 3   .....(ii)
The graph thus obtained will be as follows:

The figure thus obtained is a rectangle ABCD. The vertices of the rectangle are A(−2, 3), B(0, 3), C(0, 0) and D(−2, 0).
Length of the rectangle = AD = BC = 3 units
Breadth of the rectangle = CD = AB =  2 units
∴ Area of rectangle ABCD
=3 × 2 = 6 square units

Page No 3.32

Q.42. Draw the graphs of the pair of linear equations x − y + 2 = 0 and 4x − y − 4 = 0 . Calculate the area of the triangle formed by the lines so drawn and the x-axis .
Ans. The given linear equations are
x − y + 2 = 0                   .....(i)
4x − y − 4 = 0                 .....(ii)
For (i), we have

For (ii), we have

The graph of the lines represented by the given equations is shown below:

The triangle thus obtained has the vertices A(2, 4), B(−2, 0) and D(1, 0).
∴ Area of triangle ADB

= 6 square units
Thus, the area of the triangle formed by the given lines and the x-axis is 6 square units.

Page No 3.44

Q.1. Solve the following systems of equations:
11x + 15y + 23 = 0
7x − 2y − 20 = 0
Ans. The given equations are:
11x + 15y + 23 = 0  ....(i)
7x − 2y − 20 = 0   ....(ii)
Multiply equation (i) by 2 and equation (ii) by 15, and add both equations we get 
Put the value of x in equation (i) we get11 x 2 + 15y + 23= 0
⇒ 15y = - 45
⇒ y = -3
Hence the value of x = 2 and y = -3

Q.2. Solve the following systems of equations:
3x − 7y + 10 = 0
y − 2x − 3 = 0
Ans.2. The given equations are:
3x − 7y + 10 = 0  ....(i)
y − 2x − 3 = 0  ....(ii)
Multiply equation (i) by 2 and equation (ii)  by 3, and add both equations we get
Put the value of  y in equation (i) we get
3x - 7 x 1 + 10 = 0
⇒ 3x = -3
⇒ x = - 1
Hence the value of x = -1 and y = 1

Q.3. Solve the following systems of equations:
0.4x + 0.3y = 1.7
0.7x − 0.2y = 0.8
Ans. The given equations are:
0.4x + 0.3y = 1.7  ....(i)
0.7x − 0.2y = 0.8  ....(ii)
Multiply equation (i) by 2 and equation (ii) by 3, and add both equations we get
Put the value of x in equation (i) we get 
0.4 x 2 + 0.3y = 1.7
⇒ 0.3y = 0.9
⇒ y = 3
Hence the value of x = 2 and y = 3

Q.4. Solve the following systems of equations:

Ans. The given equations are:


Subtract (ii) from (i) we get

Put the value of x in equation (ii) we get

Hence the value of x = 0.4 and y = 0.6

Q.5. Solve the following systems of equations:
7(y + 3) − 2(x + 2) = 14
4(y − 2) + 3(x − 3) = 2
Ans. The given equations are:
7(y + 3) − 2(x + 2) = 14 ...(i)
7y - 2x = -3
4(y − 2) + 3(x − 3) = 2 ...(ii)
4y + 3x = 19
Multiply equation (i) by and equation (ii) by  and add both equations we get

Put the value of x in equation (i) we get7 x 1 - 2x = -3
⇒ - 2x = -10
⇒ x = 5
Hence the value of  x = 5 and y = 1

Q.6. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (i) by 1/3 and add both equations we get

Put the value of x in equation (i) we get

Hence the value of x = 14 and y = 9

Q.7. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (i) by 1/3 and equation (ii) by 1/4 and add both equations we get

Put the value of x in equation (i) we get

Hence the value of x = 6 and y = 36.

Q.8. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (i) by and equation (ii) by 3 and add both equations we get

Put the value of x in equation (i) we get

Hence the value of x = 2 and y = 2

Q.9. Solve the following systems of equations

Ans. The given equations are:

Multiply equation (i) by 4 and subtract equations (i) - (ii) we get
4x + 2y = 16

Put the value of x in equation (i), we get

Hence the value of x and y are x = 3 y = 2

Q.10. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (ii) by 2 and subtract equation (ii) from (i) we get

Put the value of x in equation (i), we get

Hence the value of x = 1/2 and y = 1/2.

Q.11. Solve the following systems of equations:
√2x  – √3y = 0
√3x – √8y = 0
Ans. The given equations are:

Multiply equation (i) by and equation (i) by √2 and subtract equation (ii) from (i) we get

Put the value of y in equation (i), we get
√2x + 0 x y = 0
⇒ x = 0
Hence the value of x = 0 and y = 0

Q.12. Solve the following systems of equations:

Ans. The given equations are:


Multiply equation (1) by 14 , we get
462x − 14y = 1330     ......(3)
adding (2) and (3), we get
(x + 14y) + (462x − 14y) = 59 + 1330
⇒ 463 x = 1389
⇒x = 3
Substituting the value of x in (2), we get
3 + 14y = 59
⇒ 14y = 59 − 3
⇒ 14y = 56
⇒ y = 4
Hence the value of x and y are x = 3 and y = 4

Q.13. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (i) by 3 (ii) by 2 and subtract equation (ii) from (i) we get

Put the value of y in equation (i), we get

Hence the value of x = 3 and y = - 1

Q.14. Solve the following systems of equations:
0.5x + 0.7y = 0.74
0.3x + 0.5y = 0.5
Ans. The given equations are:
0.5x + 0.7y = 0.74   ...(i)
0.3x + 0.5y = 0.5   ...(ii)
Multiply equation (i) by 0.5 and (ii) by 2 and subtract equation (ii) from (i) we get

Put the value of  x in equation (i), we get

Hence the value of x = 0.5 and y = 0.7

Page No 3.45

Q.15. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (ii) by 1/2 and add both equations we get

Put the value of x in equation (i), we get

Hence the value of

Q.16. Solve the following systems of equations:

Ans.
The given equations are:

Multiply equation (i) by   (ii)and subtract equation (ii) from (i) we get

Put the value of x in equation (i), we get

Hence the value of  and y = 1/3

Q.17. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (ii) by 2 and subtract (ii) from (i), we get

Put the value of u in equation (i), we get

Hence the value of

Q.18. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (i) by 3 and add both equations, we get

Put the value of x in equation (i), we get

Hence the value of

Q.19. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (i) by 8 and subtract from equation (i), we get

Put the value of x in equation (i), we get

Hence the value of x = 4 and y = 10

Q.20. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (i) by 3/7 and equation (ii) add both equations, we get

Put the value of x  in equation (i), we get

Hence the value of

Q.21. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (i) by 4 and equation (ii) by 3, add both equations, we get

Put the value of x in equation (i), we get

Hence the value of

Q.22. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (i) by 4 and equation (ii) by 5 and subtract (ii) from (i) we get

Put the value of x in equation (i), we get

Hence the value of

Q.23. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (i) by 4 and equation (ii) by 3 and add both equations we get

Put the value of x in equation (i), we get

Hence the value of

Q.24. Solve the following systems of equations:
Ans.
 
Ans. The given equations are:

Multiply equation (i) by 3 and add both equations we get

Put the value of x in equation (i), we get

Hence the value of x = 4 and y = 9

Q.25. Solve the following systems of equations:

Ans. The given equations are:

Adding both equations, we get

Put the value of y in equation (i), we get

Hence the value of

Q.26. Solve the following systems of equations:

Ans. The given equations are:

Multiply equation (i) by 3 and subtract (ii) from (i), we get

Put the value of x in equation (i), we get

Hence the value of x = 1 and y = 3