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Solve the system of equations by elimination.\newline−x−3y−2z=−20-x - 3y - 2z = -20\newline−2x−y+3z=−16-2x - y + 3z = -16\newlinex−2y−2z=−13x - 2y - 2z = -13

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Q. Solve the system of equations by elimination.\newline−x−3y−2z=−20-x - 3y - 2z = -20\newline−2x−y+3z=−16-2x - y + 3z = -16\newlinex−2y−2z=−13x - 2y - 2z = -13
  1. Combine Equations to Eliminate x: First, let's add the first and third equations to eliminate x.\newline(−x−3y−2z)+(x−2y−2z)=−20+(−13)(-x - 3y - 2z) + (x - 2y - 2z) = -20 + (-13)\newline−x+x−3y−2y−2z−2z=−33-x + x - 3y - 2y - 2z - 2z = -33\newline−5y−4z=−33-5y - 4z = -33
  2. Multiply and Combine Equations: Now, let's multiply the second equation by 22 so we can eliminate xx with the first equation.\newline2(−2x−y+3z)=2(−16)2(-2x - y + 3z) = 2(-16)\newline−4x−2y+6z=−32-4x - 2y + 6z = -32
  3. Solve System of Equations: Add the modified second equation to the first equation.\newline(−x−3y−2z)+(−4x−2y+6z)=−20+(−32)(-x - 3y - 2z) + (-4x - 2y + 6z) = -20 + (-32)\newline−x−4x−3y−2y−2z+6z=−52-x - 4x - 3y - 2y - 2z + 6z = -52\newline−5x−5y+4z=−52-5x - 5y + 4z = -52
  4. Eliminate z: Now, let's solve the system of two equations we have:\newline−5y−4z=−33-5y - 4z = -33\newline−5x−5y+4z=−52-5x - 5y + 4z = -52\newlineWe can add these two equations to eliminate z.\newline(−5y−4z)+(−5x−5y+4z)=−33+(−52)(-5y - 4z) + (-5x - 5y + 4z) = -33 + (-52)\newline−5y−5x−5y=−85-5y - 5x - 5y = -85\newline−5x−10y=−85-5x - 10y = -85
  5. Solve for y: Divide the last equation by −5-5 to simplify.−5x−10y=−85-5x - 10y = -85x+2y=17x + 2y = 17
  6. Correct Previous Step: Now, let's solve for yy using the equation −5y−4z=−33-5y - 4z = -33. We can substitute xx from x+2y=17x + 2y = 17 into this equation. But wait, I made a mistake in the previous step, I should have divided by −5-5 correctly. Let's correct that. −5x−10y=−85-5x - 10y = -85 x+2y=17x + 2y = 17 should be x+2y=−17x + 2y = -17

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