Systems of Equations

Solve for each variable:

-4x-6y+5z=21

3x+4y-2z=-15

-7x-5y+3z=15

$$\small{\text{$
\begin{array}{rrrrcrl}
(1) & -4x&-6y&+5z&=&21 \quad &\\
(2) & 3x&+4y&-2z&=&-15 \quad & | \quad \cdot \frac{4}{3}\\
(3) &-7x&-5y&+3z&=&15 \quad & | \quad \cdot \frac{4}{-7}\\
\\
\hline
\\
(1) & -4x&-6y&+5z&=&21 \quad &\\
(2) & 3\cdot\frac{4}{3}x &+4\cdot \frac{4}{3} y&-2\cdot \frac{4}{3} z&=&-15\cdot \frac{4}{3} \quad &\\
(3) &-7\cdot \frac{4}{-7}x&-5\cdot \frac{4}{-7}y&+3\cdot \frac{4}{-7}z&=&15\cdot \frac{4}{-7} \quad &\\
\\
\hline
\\
(1) & -4x&-6y&+5z&=&21 \quad &\\
(2) & 4x &+\frac{16}{3} y&-\frac{8}{3} z&=&-20 \quad &\\
(3) & 4x &+\frac{20}{7}y&-\frac{12}{7}z&=&-\frac{60}{7}\quad &\\
\\
\hline
\\
(1) & -4x&-6y&+5z&=&21 \quad &\\
(2)+(1) & 0 &-\frac{2}{3} y&+\frac{7}{3} z&=&1 \quad &\\
(3)+(1) & 0 &-\frac{22}{7}y&+\frac{23}{7}z&=&\frac{87}{7}\quad &\\
\\
\hline
\\
(1) & -4x&-6y&+5z&=&21 \quad &\\
(2) & 0 &-\frac{2}{3} y&+\frac{7}{3} z&=&1 \quad &\\
(3) & 0 &-\frac{22}{7}y&+\frac{23}{7}z&=&\frac{87}{7}\quad & | \quad \cdot \frac{-7}{22}\cdot \frac{2}{3}\\
\\
\hline
\\
(1) & -4x&-6y&+5z&=&21 \quad &\\
(2) & 0 &-\frac{2}{3} y&+\frac{7}{3} z&=&1 \quad &\\
(3) & 0 &-\frac{22}{7}\cdot \frac{-7}{22}\cdot\frac{2}{3}y&+\frac{23}{7} \cdot \frac{-7}{22}\cdot \frac{2}{3}z&=&\frac{87}{7} \cdot \frac{-7}{22}\cdot\frac{2}{3}\quad & \\
\\
\hline
\\
(1) & -4x&-6y&+5z&=&21 \quad &\\
(2) & 0 &-\frac{2}{3} y&+\frac{7}{3} z&=&1 \quad &\\
(3) & 0 & +\frac{2}{3}y& -\frac{23}{22} \cdot \frac{2}{3}z&=&-\frac{87}{22}\cdot \frac{2}{3}\quad & \\
\\
\hline
\\
(1) & -4x&-6y&+5z&=&21 \quad &\\
(2) & 0 &-\frac{2}{3} y&+\frac{7}{3} z&=&1 \quad &\\
(3)+(2) & 0 & 0 & \frac{108}{66}z&=&-\frac{108}{66}\quad & \\
\end{array}
$}}$$

$$\small{\text{$
\begin{array}{rcl}
\frac{108}{66}z &=& -\frac{108}{66}\\\\
z &=& \dfrac{ -\frac{108}{66} } {\frac{108}{66}} \\\\
z &=& - \frac{108}{66 }\cdot \frac{66}{108} \\\\
\boxed{\, z = -1 \, }
\end{array}
$}}$$

$$\small{\text{$
\begin{array}{rcl}
-\frac{2}{3} y+\frac{7}{3} z&=&1 \\\\
-\frac{2}{3} y+\frac{7}{3} \cdot(-1)&=&1 \\\\
-\frac{2}{3} y-\frac{7}{3} &=&1 \\\\
-\frac{2}{3} y &=&1+\frac{7}{3} \\\\
-\frac{2}{3} y &=&\frac{10}{3} \\\\
y &=& \frac{10}{3}\cdot ( -\frac{3}{2} ) \\\\
\boxed{\, y = -5 \, }
\end{array}
$}}$$

$$\small{\text{$
\begin{array}{rcl}
-4x-6y+5z&=&21 \\\\
-4x-6\cdot(-5)+5\cdot(-1)&=&21 \\\\
-4x+30-5&=&21 \\\\
-4x+25&=&21 \\\\
-4x &=&-4 \\\\
4x &=&4 \\\\
\boxed{\, x= 1 \, }
\end{array}
$}}$$

$${\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{6}}{\mathtt{\,\times\,}}{\mathtt{y}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{z}} = {\mathtt{21}} \Rightarrow {\mathtt{x}} = {\frac{\left({\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{z}}{\mathtt{\,-\,}}{\mathtt{6}}{\mathtt{\,\times\,}}{\mathtt{y}}{\mathtt{\,-\,}}{\mathtt{21}}\right)}{{\mathtt{4}}}}$$

$${\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{y}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{z}} = -{\mathtt{15}} \Rightarrow {\mathtt{x}} = {\frac{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{z}}{\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{y}}{\mathtt{\,-\,}}{\mathtt{15}}\right)}{{\mathtt{3}}}}$$

$${\mathtt{\,-\,}}{\mathtt{7}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{y}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{z}} = {\mathtt{15}} \Rightarrow {\mathtt{x}} = {\frac{\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{z}}{\mathtt{\,-\,}}{\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{y}}{\mathtt{\,-\,}}{\mathtt{15}}\right)}{{\mathtt{7}}}}$$

Thanks Heureka,

Very impressive as usual  

The teacher did not try to make that one easy!