For what value of k the following system of equation have unique solution? 2x…
2022
For what value of k the following system of equation have unique solution?
2x + 3y + 5 = 0
kx + 6y = 7
- A.
k = 4
- B.
k ≠ 4
- C.
k ≠ -4
- D.
k = ±4
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Show answer & explanation
Correct answer: B
To determine for what value of k the system of linear equations has a unique solution, we must examine the ratio of the coefficients.
Step-by-Step Solution
Standard Form: A system of two linear equations,
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
has a unique solution if and only if the ratio of the coefficients of x is not equal to the ratio of the coefficients of y. The condition is:
a1 / a2 ≠ b1 / b2
Identify Coefficients:
Given the equations:
2x + 3y + 5 = 0 (a1 = 2, b1 = 3)
kx + 6y - 7 = 0 (a2 = k, b2 = 6)
Apply the Condition:
Set up the inequality based on the ratios:
2 / k ≠ 3 / 6
Solve for k:
Simplify the ratio on the right: 3 / 6 = 1 / 2
So, 2 / k ≠ 1 / 2
Cross-multiply: k ≠ 4