Solve the system of equations by elimination. $\newline$ $2x - y - z = -4$ $\newline$ $-2x - 3y - 2z = -16$ $\newline$ $2x + y - 2z = 16$

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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

2x - y - z = -4

\newline

-2x - 3y - 2z = -16

\newline

2x + y - 2z = 16

Eliminate x by adding equations: Add the first and second equations to eliminate x. $\newline$ $(2x - y - z) + (-2x - 3y - 2z) = -4 + (-16)$ $\newline$ $2x - y - z - 2x - 3y - 2z = -20$ $\newline$ $-4y - 3z = -20$
Eliminate $x$ by adding equations: Add the first and third equations to eliminate $x$ .
$(2x - y - z) + (2x + y - 2z) = -4 + 16$
$2x - y - z + 2x + y - 2z = 12$
$4x - 3z = 12$
Solve for x: Divide the last equation by $4$ to solve for $x$ . $\newline$ $4x - 3z = 12$ $\newline$ $x - \left(\frac{3}{4}\right)z = 3$ $\newline$ $x = 3 + \left(\frac{3}{4}\right)z$
Substitute $x$ into first equation: Substitute $x = 3 + \frac{3}{4}z$ into the first equation. $\newline$ $2(3 + \frac{3}{4}z) - y - z = -4$ $\newline$ $6 + \frac{3}{2}z - y - z = -4$ $\newline$ $6 + \frac{1}{2}z - y = -4$
Isolate terms with variables: Subtract $6$ from both sides to isolate the terms with variables. $\newline$ $6 + \left(\frac{1}{2}\right)z - y - 6 = -4 - 6$ $\newline$ $\left(\frac{1}{2}\right)z - y = -10$
Get rid of fraction: Multiply the equation by $2$ to get rid of the fraction. $2\left(\frac{1}{2}z - y\right) = 2(-10)$ $z - 2y = -20$
Align y terms: Now we have a system with two equations and two variables: $\newline$ $-4y - 3z = -20$ $\newline$ $z - 2y = -20$ $\newline$ Multiply the second equation by $2$ to align the y terms. $\newline$ $2(z - 2y) = 2(-20)$ $\newline$ $2z - 4y = -40$
Eliminate y: Add the modified second equation to the first equation to eliminate y. $\newline$ $(-4y - 3z) + (2z - 4y) = -20 + (-40)$ $\newline$ $-8y - z = -60$
Solve for y: Divide the equation by $-8$ to solve for y. $\newline$ $-8y - z = -60$ $\newline$ $y + \frac{1}{8}z = 7.5$ $\newline$ $y = 7.5 - \frac{1}{8}z$
Substitute $y$ into equation: Substitute $y = 7.5 - \frac{1}{8}z$ into the equation $z - 2y = -20$ . $\newline$ $z - 2(7.5 - \frac{1}{8}z) = -20$ $\newline$ $z - 15 + \frac{1}{4}z = -20$ $\newline$ $\frac{5}{4}z = -5$ $\newline$ $z = \frac{-5}{\frac{5}{4}}$ $\newline$ $z = -4$
Find $z$ : Substitute $z = -4$ into $y = 7.5 - \frac{1}{8}z$ to find $y$ . $\newline$ $y = 7.5 - \frac{1}{8}(-4)$ $\newline$ $y = 7.5 + 0.5$ $\newline$ $y = 8$
Find $y$ : Substitute $z = -4$ into $x = 3 + \left(\frac{3}{4}\right)z$ to find $x$ . $\newline$ $x = 3 + \left(\frac{3}{4}\right)(-4)$ $\newline$ $x = 3 - 3$ $\newline$ $x = 0$