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The GMAT Sample Math question given below is from the topic Linear Equations and tests fundamental properties of a system of linear equations.

For what values of 'k' will the pair of equations 3x + 4y = 12 and kx + 12y = 30 not have a unique solution?

1. 12
2. 9
3. 3
4. 7.5
5. 2.5

The correct choice is (B) and the correct answer is 9.

A system of linear equations ax + by + c = 0 and dx + ey + g = 0 will have a unique solution if the two lines represented by the equations ax + by + c = 0 and dx + ey + g = 0 intersect at a point.

That is, if they are not parallel lines. i.e., the two lines should have different slopes.

ax + by + c = 0 and dx + ey + g = 0 will not represent two parallel lines if their slopes are different.

i.e., when

In the question given above, a = 3, b = 4, d = k and e = 12.

Therefore, or 'k' should not be equal 9 for the pair of equations to have a unique solution.

In other words, when k = 9, the system of equation will not have any solution as the two lines represented by the equations will be parallel lines.