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?

 Apr 29, 2024

If we want the system to have an infinite number of solutions, we want two congruent equations for the system. 


We have \(6a + 2b = k + 3a + mb\). We notice that if we subtract \(3a \) from both sides, the left sides of both equations match, making life much easier for us!


Subtracting \(3a \) from both sides achieves us \(3a+2b=k+mb \). In order for the equations to be equal, we want the left sides of it to match as well, meaning we can write the equation \(k+mb=2\). If m was a integer other than 0, we wouldn't have matching equations, meaning we need to have \(m = 0\). If \(m=0\), then we have \(k=2\).


Our final answer is


\(m = 0\)



Thanks! :)

 Apr 29, 2024

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