Question

Prove that the system of equations given as 2x – 3y = 7 and 4x + ky = 9, is inconsistent for k = – 6. Also, obtain the solution of the system of equations, if k = – 1.

Answer 1

Given system:

`{{:(2x - 3y = 7),(4x + ky = 9):}`

Step 1: Check inconsistency for k = –6

Write the system:

`{{:(2x - 3y = 7),(4x - 6y = 9):}`

Multiply the first equation by 2:

4x – 6y = 14

Compare with second equation:

4x – 6y = 9

Since the left sides are identical but the right sides differ (14 ≠ 9), the system has no solution.

Hence, the system is inconsistent for k = –6.

Step 2: Find solution for k = –1

System becomes:

`{{:(2x - 3y = 7),(4x - y = 9):}`

From second equation:

4x – y = 9

⟹ y = 4x – 9

Substitute into first equation:

2x – 3(4x – 9) = 7

2x – 12x + 27 = 7

–10x = 7 – 27

–10x = –20

x = 2

Find y:

y = 4(2) – 9

= 8 – 9

= –1

(x, y) = (2, –1)