For a test question, a mathematics teacher wants to find two constants a and b such that the test item "Simplify a(x + 2y) − b(2x − y)" will have an answer of -4x + 12y. What constants a and b should the teacher use?

Answer 1

a(x + 2y) − b(2x − y) = -4x + 12y
ax + 2ay - 2bx + by = -4x + 12y
x(a-2b) + y(2a + b) = -4x + 12y

so a-2b = -4
and 2a + b = 12

first one times 2 ---> 2a - 4b = -8
second one ----> 2a + b = 12
subtract them: -5b = -20
b = 4
back in first: a - 8= - 4
a = 4

check:
4(x+2y) - 4(2x-y)
= 4x + 8y - 8x + 4y
= -4x + 12y , checks!

Answer 2

awesome thanks!