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Solve the system of equations by substitution. $\newline$ $-3x - y - 3z = -11$ $\newline$ $z = 5$ $\newline$ $x - y + 3z = 19$ $\newline$ (.,)

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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

-3x - y - 3z = -11

\newline

z = 5

\newline

x - y + 3z = 19

\newline

(____.____,____)

Substitute $z = 5$ : Substitute $z = 5$ into the first equation $-3x - y - 3z = -11$ .
$-3x - y - 3\times5 = -11$
$-3x - y - 15 = -11$
$-3x - y = 4$
Substitute $z = 5$ : Substitute $z = 5$ into the second equation $x - y + 3z = 19$ .
$x - y + 3\times5 = 19$
$x - y + 15 = 19$
$x - y = 4$
Set equations equal: Notice that both equations $-3x - y = 4$ and $x - y = 4$ have the same $y$ -term. We can use this to solve for $x$ by setting the equations equal to each other. $\newline$ $-3x - y = x - y$ $\newline$ $-3x = x$ $\newline$ $-4x = 0$ $\newline$ $x = 0$
Solve for x: Substitute $x = 0$ into one of the equations to solve for $y$ . We'll use $x - y = 4$ . $\newline$ $0 - y = 4$ $\newline$ $-y = 4$ $\newline$ $y = -4$
Find $y$ and $z$ : We have already been given $z = 5$ , and we have found $x = 0$ and $y = -4$ . The solution is the ordered triple $(x, y, z)$ . $\newline$ The solution is $(0, -4, 5)$ .

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