# GMAT Maths | Linear Equation Solutions

#### GMAT Sample Questions | Condition for NO unique solution to a system of linear equations

The GMAT sample question in quant given below is a Linear Equations question and tests concepts related to types of solutions for a system of linear equations. This concept is usually tested in the GMAT as a data sufficiency question rather than as a problem solving question. A sub 600 level GMAT practice question in system of linear equations.

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

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

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#### Condition for Unique Solution to Linear Equations

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.
i.e., if the two lines are neither parallel nor coincident.
Essentially, the slopes of the two lines should be different.

#### What does that translate into?

ax + by + c = 0 and dx + ey + g = 0 will intersect at one point if their slopes are different.
Express both the equations in the standardized y = mx + c format, where 'm' is the slope of the line and 'c' is the y-intercept.

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