\( \left\{ \begin{array} { l } { x + y + z = 1 } \\ { 2 x + 3 y + 6 z = 10 \quad L 2 } \\ { 2 x - 9 y + 12 z = - 10 \quad 13 } \end{array} \right. \)

Solve the system of equations \( x+y+z=1;2x+3y+6z=10;2x-9y+12z=-10 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+y+z=1\\2x+3y+6z=10\\2x-9y+12z=-10\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=1-y-z\\2x+3y+6z=10\\2x-9y+12z=-10\end{array}\right.\) - step2: Substitute the value of \(x:\) \(\left\{ \begin{array}{l}2\left(1-y-z\right)+3y+6z=10\\2\left(1-y-z\right)-9y+12z=-10\end{array}\right.\) - step3: Simplify: \(\left\{ \begin{array}{l}2+y+4z=10\\2-11y+10z=-10\end{array}\right.\) - step4: Solve the equation: \(\left\{ \begin{array}{l}y=8-4z\\2-11y+10z=-10\end{array}\right.\) - step5: Substitute the value of \(y:\) \(2-11\left(8-4z\right)+10z=-10\) - step6: Simplify: \(-86+54z=-10\) - step7: Move the constant to the right side: \(54z=-10+86\) - step8: Add the numbers: \(54z=76\) - step9: Divide both sides: \(\frac{54z}{54}=\frac{76}{54}\) - step10: Divide the numbers: \(z=\frac{38}{27}\) - step11: Substitute the value of \(z:\) \(y=8-4\times \frac{38}{27}\) - step12: Simplify: \(y=\frac{64}{27}\) - step13: Substitute the values of \(y,z:\) \(x=1-\frac{64}{27}-\frac{38}{27}\) - step14: Simplify: \(x=-\frac{25}{9}\) - step15: Calculate: \(\left\{ \begin{array}{l}x=-\frac{25}{9}\\y=\frac{64}{27}\\z=\frac{38}{27}\end{array}\right.\) - step16: Check the solution: \(\left\{ \begin{array}{l}x=-\frac{25}{9}\\y=\frac{64}{27}\\z=\frac{38}{27}\end{array}\right.\) - step17: Rewrite: \(\left(x,y,z\right) = \left(-\frac{25}{9},\frac{64}{27},\frac{38}{27}\right)\) La solution du système d'équations est donnée par les valeurs suivantes : \( x = -\frac{25}{9} \) \( y = \frac{64}{27} \) \( z = \frac{38}{27} \) Cela signifie que les valeurs de \( x \), \( y \) et \( z \) qui satisfont simultanément les trois équations données sont \( -\frac{25}{9} \), \( \frac{64}{27} \) et \( \frac{38}{27} \) respectivement.