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, 3x-3y= 16 

(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0 

(iv) 2x – 2y – 2 = 0, 4x – 3y – 5 = 0

Answer 1

(i) x + y = 5 ⇒ x + y – 5 = 0 

2x + 2y = 10 ⇒ 2x + 2y – 10 = 0 

a₁ = 1, b₁ = 1, c₁ = -5 

a₂ = 2, b₂ = 2, c₂ = -10 

Here, 

∴ Pair of equations are consistent 

(i) x + y = 5 

y = 5 – x

x	0	2	4
y = 5 – x	5	3	1

(ii) 2x + 2y = 10
x + y = 5
y = 5 – x

x	0	2	5
y = 5 – x	5	3	0

∴ We can give any value for ‘x’, 

i.e., solutions are infinite.
∴ P (5, 0) x = 5, y = 0

(ii) x – y = 8 ⇒ x – y – 8 = 0 

3x – 3y = 16 

⇒ 3x – 3y – 16 = 0

∴ Linear equations are in consistent tent. 

∴ Algebraically it has no solution. 

Graphical representation → Parallel Lines. 

(i) x – y = 8 

-y = 8 – x 

y = -8 + x

x	8	10	9
y = -8 + x	0	2	1

(ii) 3x – 3y = 16
-3y = 16 – 3x
3y = -16 + 3x

No solution because it is inconsistent

(iii) 2x + y – 6 = 0 

4x – 2y – 4 = 0 

Here a₁ = 2, b₁ = 1, c₁ = -6 

a₂ = 4, b₂ = -2, c₂ = -4

Pair of equations are consistent. 

Algebraically both lines intersect. 

Graphical Representation : 

(i) 2x + y = 6 

y = 6 – 2x

x	0	2
y = 6 – 2x	6	2

(ii) 4x – 2y – 4 = 0
4x – 2y = 4
-2y = 4 – 4x
2y = -4 + 4x

Solution: intersecting point, P (2, 2) 

i.e., x = 2, y = 2

(iv) 2x – 2y – 2 = 0 

4x – 3y – 5 = 0 

a₁ = 2, b₁ = -2, c₁ = -2 

a₂ = 4, b₂ = -3, c₂ = -5

Pair of equations are consistent. 

∴ Algebraically both lines intersect. 

Graphical Representation : 

(i) 2x – 2y – 2 =0 

2x – 2y = 2 

-2y = 2 – 2x 

2y = -2 + 2x

∴ y =\( \frac{-2 + 2x}{2}\)

∴ y = – 1 + x

x	2	4
y = – 1 + x	1	3

(ii) 4x – 3y – 5 = 0 

4x – 3y = 5 

-3y = 5 – 4x 

3y = -5 + 4x

∴ y = \(\frac{-5 + 4x}{3}\)

x	2	5
y = \(\frac{-5 + 4x}{3}\)	1	5

Solution: P(2, 1) 

i.e., x = 2, y = 1