Algebra practice question
In the xy-plane, the system of equations 2x - 5y = 9 and ax + 15y = 4, where a is a constant, has no solution. What is the value of a?
- A. -6Correct
- B. -3
- C. 3
- D. 6
Answer: A. -6
A system of two linear equations has no solution when the coefficients are proportional but the constants are not. Since 15 = (-3)(-5), the second equation's coefficients must be -3 times the first's: a = (-3)(2) = -6. (Then -6x + 15y = 4 while -3 times the first equation gives -6x + 15y = -27, so the lines are parallel and distinct.) Choice B reports the multiplier -3 itself rather than a. Choice C uses the ratio 15 ÷ 5 = 3, ignoring the negative sign on -5y. Choice D has the correct magnitude but drops the sign, coming from 15 ÷ 5 = 3 times 2.
Why the other answers are wrong
- B. -3
- -3 is the multiplier that takes -5y to 15y, not the value of a; you still need to apply it to the x-coefficient: a = (-3)(2) = -6.
- C. 3
- You computed 15 ÷ 5 = 3 and stopped, ignoring both the negative sign on -5y and the factor of 2 from the x-coefficient.
- D. 6
- You found the right magnitude from (15 ÷ 5)(2) = 6, but dropped the sign: since -5y becomes +15y, the multiplier is -3, so a = -6.
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