# Solve the following system of equations [((1) " ", (2x + y + 3z = 12)),((2) " ",(4y – z = -7)),((3) " ",(5x + 8z = 34))]?

Oct 22, 2016

$x = 2$
$y = - 1$
$z = 3$

#### Explanation:

color(lightgreen)(( 1 ) $\textcolor{l i g h t g r e e n}{\rightarrow}$ $2 x + y + 3 z = 12$
color(lightblue)(( 2) $\textcolor{l i g h t b l u e}{\rightarrow}$ $4 y - z = - 7$
color(pink)(( 3) $\textcolor{\pi n k}{\rightarrow}$ $5 x + 8 z = 34$

Using color(lightblue)((2):
color(orange)((4) $\textcolor{\mathmr{and} a n \ge}{\rightarrow}$ $z = 4 y + 7$

Substituting the value of $z$ in color(pink)((3):
$5 x + 8 \left(4 y + 7\right) = 34$

$5 x + 32 y + 56 = 34$

color(blue)((5) $\textcolor{b l u e}{\rightarrow}$ $5 x + 32 y = - 22$

Substituting value of $z$ in color(lightgreen)((1):
$2 x + y + 3 \left(4 y + 7\right) = 12$

$2 x + y + 12 y + 21 = 12$

color(purple)((6) $\textcolor{p u r p \le}{\rightarrow}$ $2 x + 13 y = - 9$

Solving color(blue)((5) & color(purple)((6) simultaneously:
$5 x + 32 y = - 22$
$2 x + 13 y = - 9$

Multiply color(purple)((6) by $- 2.5$:
$\cancel{5 x} + 32 y = - 22$
$\cancel{- 5 x} - 32.5 y = + 22.5$

$- 0.5 y = 0.5$

$\textcolor{red}{y = - 1}$

Substituting value of $\textcolor{red}{y}$ in color(purple)((6):
$2 x + 13 \left(- 1\right) = - 9$

$2 x - 13 = - 9$

$2 x = 4$

$\textcolor{red}{x = 2}$

Substituting value of $\textcolor{red}{y}$ in color(orange)((4):
$z = 4 \left(- 1\right) + 7$

$z = - 4 + 7$

$\textcolor{red}{z = 3}$