how do you know what numbers to multiply to cancel out the x?

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Before solving this you need to multiply both the top and bottom equation so the coefficients cancel out. In this case, she chose to eliminate x's but you can eliminate y's as well. In order to eliminate the x's the numbers need to be additive inverse so the top equation is multiplied by -5 and the bottom is multiplied +4. Now the x coefficients are -20x and +20x and these are additive inverse. Hope this helps, she does a really good job explaining the problem.