i) Equation x + y = 5, 2x +2y = 10 is Consistent
ii) Equation x − y = 8, 3 x − 3y =16 is Inconsistent
iii) Equation 2x + y − 6 = 0, 4x − 2y − 4 = 0 is Consistent
vi) Equation 2x − 2y − 2 =0, 4x − 4y − 5 =0 is Inconsistent
Equation 1:
x + y = 5, 2x + 2y = 10
x + y = 5 .....(i)
y = 5 - x
After solving we get the coordinates as
x 0 3
y 5 2
plot (0, 5) and (3,5) on graph and join them to get equation x + y = 5
on graph and join them to get equation
2x + 2y = 10
2y = 10 -2x
y = (10 -2x) / 5
y = 5− x ...(iii)
After solving we get the coordinates as
x 5 2
y 0 3
Thus, there are an endless number of solutions to the consistent equation.
Equation 2:
x − y = 8 ...(i)
3 x − 3y =16 ..(ii)
x − y = 8
y = x - 8
After solving we get the coordinates as
x 8 0
y 0 -8
3 x − 3y =16
3y = 3x - 16
y = (3x - 16 ) / 3
y = x - 16/3
After solving we get the coordinates as
x 16/3 0
y 0 −16/3
When we plot the two equations on a graph, we can observe that the lines are parallel and hence uneven.
So, this is an inconsistent equation.
Equation 3:
2x + y − 6 = 0...(i)
4x − 2y − 4 = 0 ...(ii)
2x + y − 6 = 0...(i)
write it as
2x + y = 6
y = 6 - 2x
After solving we get the coordinates as
x 0 3
y 6 0
4x − 2y − 4 = 0 ...(ii)
4x − 2y = 4
2y = 4x - 4
Dividing by 2 we get
y = 2x - 2
After solving we get the coordinates as
x 1 0
y 0 −2
Thus, there are an endless number of solutions to the consistent equation.
Equation 4:
2x − 2y − 2 =0 ..(i)
4x − 4y − 5 =0 ..(ii)
Converting both equations to slope-intercept form, we get:
2x - 2y - 2 = 0
2x - 2y = 2
Dividing by 2 on the side we get
x - y = 1
y = x + 1
After solving we get the coordinates as
x 0 1
y − 1 0
4x - 4y - 5 = 0
4x - 4y = 5
4y = 5 - 4x
y = (5 - 4x) /4
y = x + 5/4
After solving we get the coordinates as
x 0 5/4
y −5/4 0
The answer is inconsistent since the two lines never cross.
So, this is an inconsistent equation.
Question:-
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(i) x + y = 5, 2x +2y = 10
(ii) x − y = 8, 3 x − 3y =16
(iii) 2x + y − 6 = 0, 4x − 2y − 4 = 0
(iv) 2x − 2y − 2 =0, 4x − 4y − 5 =0