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For certain values of k and m the system \begin{align*}
a + 2b &= -3, \\
4a + 2b &= k - a - mb
\end{align*}

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

 

2

Sophie's favorite (positive) number is a two-digit number. If she reverses the digits, the result is 36 less than her favorite number. Also, one digit is 1 less than double the other digit. What is Sophie's favorite number?

 Jun 26, 2024
 #2
avatar+814 
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For problem 1, we notice that both equations have the same coefficient on the term 2b.

 

This means that in order for the system to have infinitely many solutions, the coefficients of a must also be proportional.

 

We can rewrite the second equation as: 3a+2b=k+mb. Thus, for infinitely many solutions, we need [ \frac{4}{3} = \frac{-1}{m}. ]

 

Since the fraction on the left-hand side is positive, we know that m must be negative. Solving for m, we get m = −3​. Then k = 4/3​⋅−3 + 2​ = −2.

 

Therefore, k = -2 and m = -3.

 Jun 28, 2024

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