Math Problem Solver

\[\begin{aligned}x &= - \frac{11}{14}\\y &= \frac{47}{14}\\z &= \frac{3}{2}\end{aligned}\]

---

\[\begin{aligned}2 x + 3 y - z &= 7\\4 x - y + 5 z &= 1\\- x + 2 y + 3 z &= 12\end{aligned}\]

Eliminate \(x\) from equation 2 by multiply both sides of equation 1 by \(-2\), and adding the results to both sides of equation 2.

Eliminate \(x\) from equation 3 by multiply both sides of equation 1 by \(\frac{1}{2}\), and adding the results to both sides of equation 3.

\[\begin{aligned}2 x + 3 y - z &= 7\\- 7 y + 7 z &= -13\\\frac{7 y}{2} + \frac{5 z}{2} &= \frac{31}{2}\end{aligned}\]

Eliminate \(y\) from equation 3 by multiply both sides of equation 2 by \(\frac{1}{2}\), and adding the results to both sides of equation 3.

\[\begin{aligned}2 x + 3 y - z &= 7\\- 7 y + 7 z &= -13\\6 z &= 9\end{aligned}\]

Solve equation 3 for \(z\), and substitute the solution into the remaining equations.

\[\begin{aligned}2 x + 3 y &= \frac{17}{2}\\- 7 y &= - \frac{47}{2}\\z &= \frac{3}{2}\end{aligned}\]

Solve equation 2 for \(y\), and substitute the solution into the remaining equations.

\[\begin{aligned}2 x &= - \frac{11}{7}\\y &= \frac{47}{14}\\z &= \frac{3}{2}\end{aligned}\]

Solve equation 1 for the remaining unknown \(x\).

\[\begin{aligned}x &= - \frac{11}{14}\\y &= \frac{47}{14}\\z &= \frac{3}{2}\end{aligned}\]