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# System

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For a certain value of k, the system

3a + 4b = 7 + 2a

6a + 4b = k - 20b

has infinitely many solutions (a, b). What is k?

May 21, 2022

#1
+1768
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Simplify both equations to:

$$a + 4b = 7$$

$$6a + 24b = k$$

Note that the second equation is 6 times the first. For a system to have an infinite amount of solutions, the equations must be multiples of each other.

This means that $$k = 7 \times 6 = \color{brown}\boxed{42}$$

May 21, 2022