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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?

 Jun 29, 2024
 #1
avatar+1075 
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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! :)

 Jun 29, 2024

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