7X+5Y-3Z=16

7X+5Y-3Z=16
3X-5Y+2Z=-8
5X+3Y-7Z=0

Hello Guest!

$7X+5Y-3Z=16\\ \underline{3X-5Y+2Z=-8}\\ 10X-Z=8\ | \ \cdot (-29)\\ \color{green} -290X+29Z=-152$

$3X-5Y+2Z=-8\ |\ \cdot 3\\ 5X+3Y-7Z=0 \ |\ \cdot 5\\ 9X-15Y+6Z=-24\ \\ \underline{25X+15Y-35Z=0} \\ 34X-29Z=-24$

$\color{green}\underline{-290X+29Z=-152}\\ -256X=-176\\ \color{blue}X=\dfrac{11}{16}$

$-290X+29Z=-152\\ 29Z=-152+290X\\ 29Z=-152+290\cdot\dfrac{11}{16}\\ Z= \dfrac{-152+290\cdot\dfrac{11}{16}}{29}\\ {\color{blue}Z= \dfrac{379}{232}}\ (gerechnet \ von\ wolfram\ Alpha)$

$5X+3Y-7Z=0\\ Y=\dfrac{7Z-5X}{3}=\dfrac{7\cdot\dfrac{379}{232}-5\cdot \dfrac{11}{16}}{3}\\ \color{blue}Y=2\dfrac{2}{3}$

  !

7x+5y-3z=16
3x-5x+2z=-8
5x+3x-7z=0

$\begin{array}{|lrcll|} \hline (1) & 7x+5y-3z &=& 16 \\ (2) & 3x-5y+2z &=& -8 \\ (3) & 5x+3y-7z &=& 0 \\ \hline \end{array}$

$\begin{array}{|lrcll|} \hline (1) & 7x+5y-3z&=& 16 \\ (2) & 3x-5y+2z &=& -8 \\ (1)+(2): & 10x-z &=& 8 \\ & \mathbf{z} &=& \mathbf{10x-8} \qquad (4) \\ \hline \end{array}$

$\begin{array}{|lrcll|} \hline (2) & 3x-5y+2z &=& -8 \quad | \quad \mathbf{z=10x-8} \\ & 3x-5y+2(10x-8) &=& -8 \\ & 3x-5y+20x-16 &=& -8 \\ & 23x-5y &=& 8 \\ & 5y &=& 23x-8 \\ & \mathbf{y} &=& \mathbf{\dfrac{23x-8}{5}}\qquad (5) \\ \hline \end{array}$

$\begin{array}{|lrcll|} \hline (3) & 5x+3y-7z &=& 0 \quad | \quad \mathbf{z=10x-8} \\ & 5x+3y-7(10x-8) &=& 0 \\ & 5x+3y-70x+56 &=& 0 \\ & \mathbf{65x-3y} &=& \mathbf{56} \quad | \quad \mathbf{y=\dfrac{23x-8}{5}} \\ & 65x-3\left(\dfrac{23x-8}{5}\right) &=& 56 \quad | \quad *5 \\ & 5*65x-3*(23x-8) &=& 56*5 \\ & 325x-69x +24 &=& 280 \\ & 256x &=& 265 \\ & \mathbf{x} &=& \mathbf{1} \\ \hline \end{array}$

$\begin{array}{|lrcll|} \hline (5) & \mathbf{y} &=& \mathbf{\dfrac{23x-8}{5}} \quad | \quad \mathbf{x=1} \\ & y &=& \dfrac{23-8}{5} \\ &y &=& \dfrac{15}{5} \\ & \mathbf{y} &=& \mathbf{3} \\ \hline \end{array}$

$\begin{array}{|lrcll|} \hline (4) & \mathbf{z} &=& \mathbf{10x-8} \quad | \quad \mathbf{x=1} \\ & z &=& 10-8 \\ & \mathbf{z} &=& \mathbf{2} \\ \hline \end{array}$