+1222 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?
+1908 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! :)
