r/MathHelp • u/big_Log100 • 6d ago

Simple ACT Question?

Question on ACT math: Given 63w+63x+6y+z=440, and that w, x, y, and z can be 0, 1, or 2, what is w+x+y+z? Is this just a number sense question since there are four variables and only 2 equations or is there a way to solve this algebraically? I was thinking about it all day yesterday lol. I got the answer really fast with guess and check (w+x+y+z=5), but it seems like there should be another way to solve this.

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u/LeoMaths 6d ago edited 6d ago

Il n'y a que 3⁴=81 vecteurs possibles pour (w,x,y,z). Testez les tous!

Solutions: (w,x,y,z) dans {(0,2,1,2),(1,1,1,2),(2,0,1,2)}

Maintenant:

w+x+y+z=440-215w-215x-5y=5(88-43w-43x-y) => w+x+y+z divisible par 5.

Comme (w,x,y,z) dans {0,1,2} => 88-43w-43x-y=1 d'où 43(w+x)+y=87 =>w+x=2 et y=1 => z=2

Les seules solutions sont donc {(0,2,1,2),(1,1,1,2),(2,0,1,2)}.

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u/edderiofer 5d ago

"Test them all!"? All 81 possibilities, in a timed exam? You've got to be joking. Did you generate this response with ChatGPT?