Question

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

\begin{align*}
3a + 2b &= 2, \\
6a + 2b &= k - 3a - mb
\end{align*}has infinitely many solutions $(a,b).$ What are $k$ and $m?$

Answer 1

To find values of k and m that result in infinitely many solutions for the given system of equations, we need to set the two equations equal to each other and simplify. The value of k must be 6, and m can be any value.

Step-by-step explanation:

To find the values of k and m that result in infinitely many solutions for the system of equations, we need to set the two equations equal to each other and simplify.

1. Start by rearranging the second equation to isolate b: 6a + 2b = k - 3a - mb becomes 9a + mb + 2b = k.
2. Now, set the first equation equal to the rearranged second equation: 3a + 2b = 9a + mb + 2b.
3. Combine like terms: 6a = mb.

This equation tells us that k must equal 6 for infinitely many solutions, and m can be any value.