3. \( \left\{\begin{array}{c}x+y-z=-2 \\ x-y=5 \\ 2 x-y+3 z=10\end{array}\right. \)

Solve the system of equations \( x+y-z=-2;x-y=5;2x-y+3z=10 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+y-z=-2\\x-y=5\\2x-y+3z=10\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=-2-y+z\\x-y=5\\2x-y+3z=10\end{array}\right.\) - step2: Substitute the value of \(x:\) \(\left\{ \begin{array}{l}-2-y+z-y=5\\2\left(-2-y+z\right)-y+3z=10\end{array}\right.\) - step3: Simplify: \(\left\{ \begin{array}{l}-2-2y+z=5\\-4-3y+5z=10\end{array}\right.\) - step4: Solve the equation: \(\left\{ \begin{array}{l}z=7+2y\\-4-3y+5z=10\end{array}\right.\) - step5: Substitute the value of \(z:\) \(-4-3y+5\left(7+2y\right)=10\) - step6: Simplify: \(31+7y=10\) - step7: Move the constant to the right side: \(7y=10-31\) - step8: Subtract the numbers: \(7y=-21\) - step9: Divide both sides: \(\frac{7y}{7}=\frac{-21}{7}\) - step10: Divide the numbers: \(y=-3\) - step11: Substitute the value of \(y:\) \(z=7+2\left(-3\right)\) - step12: Simplify: \(z=1\) - step13: Substitute the values of \(y,z:\) \(x=-2-\left(-3\right)+1\) - step14: Simplify: \(x=2\) - step15: Calculate: \(\left\{ \begin{array}{l}x=2\\y=-3\\z=1\end{array}\right.\) - step16: Check the solution: \(\left\{ \begin{array}{l}x=2\\y=-3\\z=1\end{array}\right.\) - step17: Rewrite: \(\left(x,y,z\right) = \left(2,-3,1\right)\) The solution to the system of equations is \( x = 2, y = -3, z = 1 \).