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Solve the system of equations by elimination. $\newline$ $-x - y - z = 8$ $\newline$ $x + 2y - 3z = 13$ $\newline$ $3x + 2y + 2z = -20$

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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

-x - y - z = 8

\newline

x + 2y - 3z = 13

\newline

3x + 2y + 2z = -20

Combine Equations to Eliminate x: Add the first and second equations to eliminate x. $\newline$ $(-x - y - z) + (x + 2y - 3z) = 8 + 13$ $\newline$ $-y - z + 2y - 3z = 21$ $\newline$ $y - 4z = 21$
Multiply and Add for x Elimination: Multiply the first equation by $3$ and add it to the third equation to eliminate $x$ . $\newline$ $3(-x - y - z) + (3x + 2y + 2z) = 3(8) + (-20)$ $\newline$ $-3x - 3y - 3z + 3x + 2y + 2z = 24 - 20$ $\newline$ $-y - z = 4$
Eliminate y: Now we have two new equations: $\newline$ $y - 4z = 21$ $\newline$ $-y - z = 4$ $\newline$ Add these two equations to eliminate y. $\newline$ $(y - 4z) + (-y - z) = 21 + 4$ $\newline$ $-5z = 25$ $\newline$ $z = -5$
Find $y$ : Substitute $z = -5$ into $-y - z = 4$ to find $y$ .
$-y - (-5) = 4$
$-y + 5 = 4$
$-y = -1$
$y = 1$
Find $x$ : Substitute $y = 1$ and $z = -5$ into the first original equation to find $x$ .
$-x - y - z = 8$
$-x - 1 - (-5) = 8$
$-x + 4 = 8$
$-x = 4$
$x = -4$

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