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Solve the system of equations by substitution. $\newline$ $x = 7$ $\newline$ $2x - y - 3z = -11$ $\newline$ $-2x + 2y + 3z = 18$

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Solve a system of equations in three variables using substitution

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Q. Solve the system of equations by substitution.

\newline

x = 7

\newline

2x - y - 3z = -11

\newline

-2x + 2y + 3z = 18

Substitute $x = 7$ : Substitute $x = 7$ into the second equation $2x - y - 3z = -11$ . $\newline$ $2(7) - y - 3z = -11$ $\newline$ $14 - y - 3z = -11$
Isolate y: Now, let's isolate y in the equation. $\newline$ $-y - 3z = -11 - 14$ $\newline$ $-y - 3z = -25$ $\newline$ $y = 25 + 3z$
Substitute $x = 7$ : Substitute $x = 7$ into the third equation $-2x + 2y + 3z = 18$ . $\newline$ $-2(7) + 2y + 3z = 18$ $\newline$ $-14 + 2y + 3z = 18$
Isolate $2y + 3z$ : Now, let's isolate $2y + 3z$ in the equation. $\newline$ $2y + 3z = 18 + 14$ $\newline$ $2y + 3z = 32$
Substitute $y = 25 + 3z$ : Substitute $y = 25 + 3z$ into $2y + 3z = 32$ . $2(25 + 3z) + 3z = 32$ $50 + 6z + 3z = 32$
Combine like terms: Combine like terms. $\newline$ $9z = 32 - 50$ $\newline$ $9z = -18$
Divide by $9$ : Divide both sides by $9$ to solve for $z$ . $z = \frac{-18}{9}$ $z = -2$
Substitute $z = -2$ : Substitute $z = -2$ into $y = 25 + 3z$ to find $y$ .
$y = 25 + 3(-2)$
$y = 25 - 6$
$y = 19$
Find $y$ : We have found the values for $x$ , $y$ , and $z$ . $x = 7$ , $y = 19$ , $z = -2$

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