For certain values of $k$ and $m,$ the system

a + 2b = -3 - 7a + b,

4a + 2b = k - 5a - mb

has infinitely many solutions $(a,b).$ What are $k$ and $m?$

magenta Jul 31, 2024

#1**+1 **

Let's simplify the two equations a bit first. We get that

\(8a + b = -3 \\ 9a + (2 +m)b = k \)

Now,

This will have infinte solutions when the equations are the same...so....multiply the first equation by 9 and the second by 8 and we get

\(72a + 9b = -27 \\\ 72a + 8 (2 + m)b = 8k\)

Thus, we can no easily solve for m and k. Solving for m, we get

\(9 = 8(2 + m) \\ 9 = 16 + 8m \\ -7 = 8m \\ m = -7/8\)

Solving for k, we get

\(8 k = -27 \\ k = -27/8 \)

Thanks! :)

NotThatSmart Jul 31, 2024