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For certain values of k and m, the system

3a + 2b = 2

6a + 2b = k - 3a - mb

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

 Nov 11, 2023
edited by iuuhfeusedhu  Nov 11, 2023
 #1
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For the system to have infinitely many solutions, the left-hand sides must be equal to each other, regardless of the values of a and b. Thus, 3a + 2b = 6a + 2b, which implies a = 0. Substituting this value into the first equation yields 2b = 2. Hence, b = 1. Then, the second equation becomes 0 - mb = k - 3(0) - m(1), which simplifies to k = m.

Therefore, k = m = \boxed{1}.

 Nov 11, 2023
 #2
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[1] 3a + 2b = 2

 

6a + 2b = k - 3a - ab

 

[2] 9a + (2 + m) * b = k

Equation [1] is equivalent to

[3] 9a + 6b = 6

 

There are an infinite number of solutions if [2] and [3] are equivalent. That is true only if the coefficients of b in both equations are the same and the constants in both are the same.

2+m = 6 --> m = 4
k = 6

ANSWERS: k = 6; m = 4
 

 Nov 14, 2023

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