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Solve the system of equations by elimination. $\newline$ $-3x + y - 2z = 3$ $\newline$ $2x - y + 3z = -18$ $\newline$ $x + y + 2z = -9$

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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

-3x + y - 2z = 3

\newline

2x - y + 3z = -18

\newline

x + y + 2z = -9

Combine Equations to Eliminate $y$ : Combine the first and second equations to eliminate $y$ .
$(-3x + y - 2z) + (2x - y + 3z) = 3 + (-18)$
$-3x + y - 2z + 2x - y + 3z = -15$
$-x + z = -15$
Combine Equations to Eliminate $y$ : Combine the second and third equations to eliminate $y$ .
$(2x - y + 3z) + (x + y + 2z) = -18 + (-9)$
$2x - y + 3z + x + y + 2z = -27$
$3x + 5z = -27$
Multiply to Eliminate $x$ : Multiply the equation $-x + z = -15$ by $3$ to help eliminate $x$ when added to $3x + 5z = -27$ . $\newline$ $3(-x + z) = 3(-15)$ $\newline$ $-3x + 3z = -45$
Add Equations to Eliminate $x$ : Add the equations $-3x + 3z = -45$ and $3x + 5z = -27$ to eliminate $x$ .
$(-3x + 3z) + (3x + 5z) = -45 + (-27)$
$-3x + 3z + 3x + 5z = -72$
$8z = -72$
Find $z$ : Divide both sides by $8$ to find $z$ . $\frac{8z}{8} = \frac{-72}{8}$ $z = -9$
Find $x$ : Substitute $z = -9$ into $-x + z = -15$ to find $x$ . $\newline$ $-x + (-9) = -15$ $\newline$ $-x - 9 = -15$
Solve for x: Add $9$ to both sides to solve for x. $\newline$ $-x - 9 + 9 = -15 + 9$ $\newline$ $-x = -6$ $\newline$ $x = 6$
Find $y$ : Substitute $x = 6$ and $z = -9$ into the third equation $x + y + 2z = -9$ to find $y$ . $6 + y + 2(-9) = -9$ $6 + y - 18 = -9$
Solve for y: Add $18$ to both sides and then subtract $6$ to solve for $y$ . $\newline$ $y - 18 + 18 = -9 + 18 - 6$ $\newline$ $y = 3$

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