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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Explanatory Answer

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.

ax + by + c = 0 can be written as y = \-\frac{a}{b}x -\frac{c}{a} \\)
And dx + ey + g = 0 can be written as y = \-\frac{d}{e}x -\frac{g}{e} \\)
Slope of the first line is \-\frac{a}{b} \\) and that of the second line is \-\frac{d}{e} \\)
For a unique solution, the slopes of the lines should be different.
∴ \-\frac{a}{b} \neq -\frac{d}{e} \\)
Or \\frac{a}{d} \neq \frac{b}{e} \\)

Condition for the equations to NOT have a unique solution

The slopes should be equal
Or \\frac{a}{d} = \frac{b}{e} \\)

Apply the condition in the given equations to find k

In the question given above, a = 3, b = 4, d = k and e = 12.
Therefore, \\frac{3}{k} = \frac{4}{12} \\)
Or 'k' should be equal to 9 for the system of linear equations to NOT have a unique solution.

The question is "What is the value of k?
When k = 9, the system of equations will represent a pair of parallel lines (their y-intercepts are different). So, there will be NO solution to this system of linear equations in two variables.

Choice A is the correct answer.


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