RD Chapter 3 Pair of Linear Equations in Two Variables Ex 3.2 Solutions
Question - 11 : - Show graphically that each one of the following systems of equations has infinitely many solutions :2x + 3y = 6
4x + 6y = 12 [CBSE2010]
Answer - 11 : -
2x + 3y = 6 ……….(i)
4x + 6y = 12 ……….(ii)
2x = 6 – 3y
Now plot the points of both lines on the graph and join them, we see that all the points lie on the same straight line
This system has infinitely many solutions
Question - 12 : - x – 2y = 5
3x – 6y = 15
Answer - 12 : -
x – 2y = 5
x = 5 + 2y
Substituting some different values of y, we get corresponding values of x as shown below:
Now plot these points on the graph and join them
3x – 6y = 15
=> 3x = 15 + 6y
x =
Substituting some different values of y, we get corresponding values of x as shown below:
Now plot there points on the graph and join then
We see that these two lines coincide each other
This system has infinitely many solutions.
Question - 13 : - 3x +y = 8
6x + 2y = 16
Answer - 13 : -
3x + y = 8 => y = 8 – 3x
Substituting some different values of x, we get corresponding values of y as shown below:
Now plot these points on the graph and join them
6x + 2y – 16 => 6x = 16 – 2y
Substituting some different values of y, we get their corresponding values of x as shown below:
Now plot the points and point them
We see that the two lines coincide each other
This system has infinitely many solutions
Question - 14 : - x- 2y + 11 = 0
3x – 6y + 33 = 0
Answer - 14 : -
x – 2y + 11 = 0
x = 2y – 11
Substituting some different values of y, we get their corresponding values of x as shown below
Plot the points on the graph and join them 3x – 6y + 33 = 0
3x = 6y – 33
x =
Substituting some different values of y, we get corresponding values of x as shown below
Plot the points on the graph and join them we see that the two lines coincide each other
This system has infinitely many solutions.
Question - 15 : - 3x – 5y = 20
6x – 10y = -40 (C.B.S.E. 1995C)
Answer - 15 : -
3x – 5y = 20
3x = 20 + 5y
x =
Substituting some different values of y, we get their corresponding values of x as shown below
Plot the points on the graph and join them
6x – 10y = -40
6x = 10y – 40
x =
Substituting some different values of y, we get their corresponding values of x as shown below
Plot the points on the graph and join them we see that the lines are parallel
The given system of equations is inconsistant and has no solution.
Question - 16 : - 2y – x = 9
6y – 3x = 21 (C.B.S.E. 1995C)
Answer - 16 : -
2y – x = 9
=> x = 2y – 9
Substituting some different values of y, we get their corresponding values of x as shown below:
Now plot the points on the graph and join them
6y – 3x = 21
=> 6y = 21 + 3x
y =
Substituting some different values of x, we get their corresponding values of y as shown below:
Now plot the points on the graph and join them we see that the lines are parallel
The system of equations is inconsistant and therefore has no solution.
Question - 17 : - 3x – 4y – 1 = 0
2x – (8/3) y + 5 = 0
Answer - 17 : -
3x – 4y -1 = 0
3x = 4y + 1
x =
Substituting some different values of y, we get their corresponding values of x as shown below:
Now plot the points on the graph and join them
2x – (8/3) y + 5 = 0
=> 6x – 8y + 15 = 0
=> 6x = 8y – 15
=> x =
Now substituting some different values of y, we get their corresponding values of x as shown below
Plot the points on the graph and join them We see that the lines are parallel
The system of equations is inconsistant Therefore has no solution.
Question - 18 : - Determine graphically the vertices of the triangle the equations of whose sides are given below :
(i) 2y – x = 8, 5y – x = 14 and y – 2x = 1 (C.B.S.E. 1994)
(ii) y = x, y = 0 and 3x + 3y = 10 (C.B.S.E. 1994)
Answer - 18 : -
(i) Equations of the sides of a triangle are 2y – x = 8, 5y – x = 14 and y – 2x = 1
2y – x = 8
x = 2y – 8
Substituting some different values of y, we get their corresponding values of x as shown below
Now plot the points and join them Similarly in 5y – x = 14
x = 5y – 14
Now plot these points and join them in each case
We see that these lines intersect at (-4, 2), (1, 3) and (2, 5) which are the vertices of the triangle so formed.
(ii) y = x, y = 0 and 3x + 3y = 10
y = x
Substituting some different values of x, we get their corresponding values of y, as shown below
Answer - 19 : -
x – 2y = 2
x = 2y + 2
Substituting some values of y, we get their corresponding values of x, as shown below
Now plot the points on the graph and join them
4x – 2y = 5
4x = 2y + 5
x =
Substituting some different values of y, we get their corresponding values bf x as shown below
Now plot the above points and join them We see that there two lines intersect each other
The system is consistant
Question - 20 : - Determine by drawing graphs, whether the following system of linear equations has a unique solution or not :
(i) 2x – 3y = 6, x + y = 1 (C.B.S.E. 1994)
(ii) 2y = 4x – 6, 2x = y + 3 (C.B.S.E. 1995C)
Answer - 20 : -
(i) 2x – 3y = 6, x + y = 1
2x – 3y = 6
=> 2x = 6 + 3y
=> x =
Substituting some different values of y, we get their coiTesponding values of x show below
Now plot the points and join them
x + y = 1 => x = 1 – y
Substituting some different of y, we get their corresponding value of x as given below
Now plot the points on the graph and join them we see that the lines intersect at a point
This system has a unique solution.
(ii) 2y = 4x – 6, 2x = y + 3
2y = 4x – 6
y =
= 2x – 3Substituting some different values of x, we get their corresponding values of y as shown below
Now plot the points on the graph and join them
2x = y + 3
x =
Substituting some different values of y, we get their corresponding values of x as shown below
Plot the points on the graph and join them We see the lines coinside each other
This system has no unique solution.