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Solve the system of equations by elimination. $\newline$ $-2x - 2y + z = 9$ $\newline$ $-2x - 2y - z = 15$ $\newline$ $-3x - y + 2z = 12$

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Solve a system of equations in three variables using elimination

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Q. Solve the system of equations by elimination.

\newline

-2x - 2y + z = 9

\newline

-2x - 2y - z = 15

\newline

-3x - y + 2z = 12

Add Equations to Eliminate z: Step $1$ : Add the first two equations to eliminate z. $\newline$ $-2x - 2y + z = 9$ $\newline$ $-2x - 2y - z = 15$ $\newline$ $(-2x - 2y + z) + (-2x - 2y - z) = 9 + 15$ $\newline$ $-4x - 4y = 24$ $\newline$ Divide by $-4$ to simplify. $\newline$ $x + y = -6$
Multiply Third Equation: Step $2$ : Multiply the third equation by $2$ to prepare for elimination with the first equation. $\newline$ $-3x - y + 2z = 12$ $\newline$ $2(-3x - y + 2z) = 2(12)$ $\newline$ $-6x - 2y + 4z = 24$
Add Modified Third Equation: Step $3$ : Add the modified third equation to the first equation to eliminate $y$ . $-2x - 2y + z = 9$ $-6x - 2y + 4z = 24$ $(-2x - 2y + z) + (-6x - 2y + 4z) = 9 + 24$ $-8x + 5z = 33$
Solve for z: Step $4$ : Solve for z using the equation from Step $3$ . $\newline$ $-8x + 5z = 33$ $\newline$ Substitute $x$ from $x + y = -6$ into this equation. $\newline$ $-8(-6 - y) + 5z = 33$ $\newline$ $48 + 8y + 5z = 33$ $\newline$ $5z = 33 - 48 - 8y$ $\newline$ $5z = -15 - 8y$ $\newline$ $z = -3 - 1.6y$

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