For certain values of k and m, the system

3a + 2b = 2

a + 2b = k - 8a - mb

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

learnmgcat Jun 29, 2024

#1**+1 **

The only way for the system to have infinitely solutions is where we have two exactly congruent equations.

We first notice that we have

\(6a + 2b = k + 3a + mb\)

Subtracting 3a from both sides will achieve us the same exact left sides of the equations. We have

\(3a+2b=2\\ 3a+2b = k+mb\)

Setting the right sides of the equation to equal each other, we have

\(k+mb=2\)

Now, let's note that if m was any number other than 0, the two equations would fail to match each other. Therefore, m must be 0.

For this value of m, we have k is 2. So we have

\(m=0\\ k=2\)

So our final answer is m = 0 and k = 2.

Thanks! :)

NotThatSmart Jun 29, 2024