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Solve the system of equations by elimination. $\newline$ $x - 3y - 2z = 10$ $\newline$ $3x + 2y + 2z = 14$ $\newline$ $2x - 3y - 2z = 16$ $\newline$ $(\_,\_,\_)$

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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 - 3y - 2z = 10

\newline

3x + 2y + 2z = 14

\newline

2x - 3y - 2z = 16

\newline

(\_,\_,\_)

Combine equations for z elimination: Add the first and third equations to eliminate z. $\newline$ $(x - 3y - 2z) + (2x - 3y - 2z) = 10 + 16$ $\newline$ $3x - 6y = 26$
Double second equation: Now, let's double the second equation to prepare it for elimination with the third equation. $\newline$ $2(3x + 2y + 2z) = 2(14)$ $\newline$ $6x + 4y + 4z = 28$
Add equations to eliminate z: Add the new equation from the previous step to the third equation to eliminate z. $\newline$ $(6x + 4y + 4z) + (2x - 3y - 2z) = 28 + 16$ $\newline$ $8x + y = 44$
Prepare for y elimination: Now we have two equations with just x and y: $\newline$ $3x - 6y = 26$ $\newline$ $8x + y = 44$ $\newline$ Let's multiply the second equation by $6$ to prepare it for elimination with the first. $\newline$ $6(8x + y) = 6(44)$ $\newline$ $48x + 6y = 264$
Add equations to eliminate $y$ : Add the new equation from the previous step to the first equation to eliminate $y$ . $(3x - 6y) + (48x + 6y) = 26 + 264$ $51x = 290$
Solve for x: Divide both sides by $51$ to find $x$ . $\newline$ $x = \frac{290}{51}$ $\newline$ $x = 5.686274509803922$

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