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Solve the system of equations by elimination. $\newline$ $-2x - 3y + z = 18$ $\newline$ $-x + 2y - 2z = -17$ $\newline$ $-2x + y - 2z = -13$

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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 - 3y + z = 18

\newline

-x + 2y - 2z = -17

\newline

-2x + y - 2z = -13

Add Equations to Eliminate y: First, let's add the first and third equations to eliminate y. $\newline$ $-2x - 3y + z + (-2x + y - 2z) = 18 + (-13)$ $\newline$ $-4x - 2y - z = 5$
Multiply and Add Equations: Now, let's multiply the second equation by $2$ so we can add it to the first equation and eliminate $y$ .
$-2(-x + 2y - 2z) = -2(-17)$
$2x - 4y + 4z = 34$
Add Modified Equations: Add the modified second equation to the first equation. $\newline$ $(-2x - 3y + z) + (2x - 4y + 4z) = 18 + 34$ $\newline$ $-7y + 5z = 52$
Solve for z: Now we have two new equations: $\newline$ $-4x - 2y - z = 5$ $\newline$ $-7y + 5z = 52$ $\newline$ Let's solve for z by multiplying the first new equation by $5$ and the second new equation by $1$ so we can add them and eliminate y. $\newline$ $5(-4x - 2y - z) = 5(5)$ $\newline$ $-7y + 5z = 52$
Add Equations to Eliminate y: After multiplying we get: $\newline$ $-20x - 10y - 5z = 25$ $\newline$ $-7y + 5z = 52$ $\newline$ Now, let's add these two equations. $\newline$ $(-20x - 10y - 5z) + (-7y + 5z) = 25 + 52$ $\newline$ $-20x - 17y = 77$

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