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Difficulty: HardSystems of Linear Equations

Consider the system of linear equations below, where kk is a constant:

ky−3x=44y−(k−1)x=3\begin{aligned} ky - 3x &= 4 \\ 4y - (k-1)x &= 3 \end{aligned}

If the system of equations has no solution, what is the sum of all possible values of kk?

  1. 1Answer
  2. B
    -1
  3. C
    4
  4. D
    7

Answer

1
For the system of equations to have no solution, the lines must be parallel. In slope-intercept form, the equations are y=3kx+4ky = \frac{3}{k}x + \frac{4}{k} and y=k−14x+34y = \frac{k-1}{4}x + \frac{3}{4}. Setting the slopes equal gives 3k=k−14\frac{3}{k} = \frac{k-1}{4}, which simplifies to the quadratic equation k2−k−12=0k^2 - k - 12 = 0. Factoring this equation yields (k−4)(k+3)=0(k-4)(k+3) = 0, giving k=4k = 4 and k=−3k = -3. Since both values yield different y-intercepts for the two lines, they both result in parallel lines with no intersection. The sum of these values is 4+(−3)=14 + (-3) = 1.

Step-by-Step Solution

1
Express both equations in slope-intercept form (y=mx+by = mx + b) to find their slopes.
The first equation becomes y=3kx+4ky = \frac{3}{k}x + \frac{4}{k} (for k≠0k \neq 0). The second equation becomes y=k−14x+34y = \frac{k-1}{4}x + \frac{3}{4}.
For a system of linear equations to have no solution, the lines must be parallel, meaning they have the same slope but different y-intercepts.
2
Set the slopes of the two lines equal to each other to solve for kk.
3k=k−14  ⟹  k(k−1)=12  ⟹  k2−k−12=0\frac{3}{k} = \frac{k-1}{4} \implies k(k-1) = 12 \implies k^2 - k - 12 = 0.
Equating the slopes allows us to find the values of kk that make the lines parallel.
3
Solve the quadratic equation k2−k−12=0k^2 - k - 12 = 0 by factoring.
(k−4)(k+3)=0  ⟹  k=4(k-4)(k+3) = 0 \implies k = 4 or k=−3k = -3.
This determines the specific values of kk that make the slopes equal.
4
Verify that both values of kk produce different y-intercepts (so the lines do not coincide) and calculate their sum.
For k=4k = 4, the y-intercepts are 44=1\frac{4}{4} = 1 and 34\frac{3}{4}, which are different. For k=−3k = -3, the y-intercepts are −43-\frac{4}{3} and 34\frac{3}{4}, which are also different. The sum of the values of kk is 4+(−3)=14 + (-3) = 1.
We must confirm the lines are parallel and not identical, then compute the required sum.

Key Concept

Systems of linear equations with no solution represent parallel lines with equal slopes and unequal y-intercepts.
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Topics

Systems of Linear Equations

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