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What is the value of x + y + z when 6x + 5y − 2z = −4 and 4x + 5y + 12z = 54?

 May 16, 2020
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You can subtract the second equation from the first to get (4x-6x) + (5y-5y) + (12z-(-2z)) = 54-(-4)
This turns into -2x +14z = 58. Dividing both sides by 2 gets you 7z-x = 29. Moving the x to the other side gets you x = 7z-29

Substituting into the second equation gives 4(7z-29)+5y+12z = 54
Expanding gives 28z-116 + 5y +12z. Combining like terms gives 40z +5y = 170. Then dividing by 5 gives 8z +y = 34. If you tried to substitute into the first equation would give you 0 = 0 which is obviously useless. However adding x-7z and 8z+y gives you x+y+(8z-7z) which is just x+y+z. We know x-7z+ 8z+y is -29 +34  = 5.
 

 May 16, 2020

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