Solve the system of equations by elimination. $\newline$ $-3x + 2y - z = -16$ $\newline$ $-3x + y - 2z = -20$ $\newline$ $x + 2y - 2z = 6$

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Algebra 2

Solve a system of equations in three variables using elimination

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Q. Solve the system of equations by elimination.

\newline

-3x + 2y - z = -16

\newline

-3x + y - 2z = -20

\newline

x + 2y - 2z = 6

Add Equations for Elimination: question_prompt: Solve the system of equations by elimination and find the values of $x$ , $y$ , and $z$ .
Multiply Third Equation: Step $1$ : Add the first and second equations to eliminate $x$ . $(-3x + 2y - z) + (-3x + y - 2z) = -16 + (-20)$ $-6x + 3y - 3z = -36$
Add Equations for Elimination: Step $2$ : Multiply the third equation by $3$ to prepare for elimination with the first equation. $\newline$ $3(x + 2y - 2z) = 3(6)$ $\newline$ $3x + 6y - 6z = 18$
Multiply Equation for Elimination: Step $3$ : Add the first and the new third equation to eliminate $x$ . $(-3x + 2y - z) + (3x + 6y - 6z) = -16 + 18$ $8y - 7z = 2$
Add Equations for Elimination: Step $4$ : Multiply the new equation from Step $1$ by $2$ to prepare for elimination with the new equation from Step $3$ . $\newline$ $2(-6x + 3y - 3z) = 2(-36)$ $\newline$ $-12x + 6y - 6z = -72$
Divide Equation for Simplification: Step $5$ : Add the new equation from Step $4$ and the new third equation to eliminate $x$ . $(-12x + 6y - 6z) + (3x + 6y - 6z) = -72 + 18$ $-9x + 12y - 12z = -54$
Add Equations for Elimination: Step $6$ : Divide the new equation from Step $5$ by $-3$ to simplify. $\newline$ $-\frac{9x}{-3} + \frac{12y}{-3} - \frac{12z}{-3} = \frac{-54}{-3}$ $\newline$ $3x - 4y + 4z = 18$
Multiply Equation for Elimination: Step $7$ : Add the new equation from Step $6$ and the second original equation to eliminate $x$ . $(3x - 4y + 4z) + (-3x + y - 2z) = 18 + (-20)$ $-3y + 2z = -2$
Add Equations for Elimination: Step $8$ : Multiply the new equation from Step $3$ by $3$ to prepare for elimination with the new equation from Step $7$ . $\newline$ $3(8y - 7z) = 3(2)$ $\newline$ $24y - 21z = 6$
Add Equations for Elimination: Step $8$ : Multiply the new equation from Step $3$ by $3$ to prepare for elimination with the new equation from Step $7$ . $\newline$ $3(8y - 7z) = 3(2)$ $\newline$ $24y - 21z = 6$ Step $9$ : Add the new equation from Step $7$ and the new equation from Step $8$ to eliminate $z$ . $\newline$ $(-3y + 2z) + (24y - 21z) = -2 + 6$ $\newline$ $21y - 19z = 4$