Math Problem Solver

The system of equations has one solution.\[x = 0\] \[y = 0\] \[z = 0\]

Solve by substitution:

Solve \(x - y + z = 0\) for \(x\). \[x = y - z\]

Substitute \(y - z\) for \(x\) in each of the remaining equations and simplify. $$\begin{aligned}x + y + 2 z &= 0 \\ \left(y - z\right) + y + 2 z &= 0 \\ z &= - 2 y \end{aligned}$$$$\begin{aligned}2 x + 3 y - 5 z &= 0 \\ 2 \left(y - z\right) + 3 y - 5 z &= 0 \\ 5 y &= 7 z \end{aligned}$$$$\begin{aligned}2 x - y + z &= 0 \\ 2 \left(y - z\right) - y + z &= 0 \\ y &= z \end{aligned}$$$$\begin{aligned}4 x + 3 y - z &= 0 \\ 4 \left(y - z\right) + 3 y - z &= 0 \\ 7 y &= 5 z \end{aligned}$$

Solve \(z = - 2 y\) for \(y\). \[y = - \frac{z}{2}\]

Substitute \(- \frac{z}{2}\) for \(y\) in each of the remaining equations and simplify. $$\begin{aligned}5 y &= 7 z \\ 5 \left(- \frac{z}{2}\right) &= 7 z \\ z &= 0 \end{aligned}$$$$\begin{aligned}y &= z \\ \left(- \frac{z}{2}\right) &= z \\ z &= 0 \end{aligned}$$$$\begin{aligned}7 y &= 5 z \\ 7 \left(- \frac{z}{2}\right) &= 5 z \\ z &= 0 \end{aligned}$$

Use substitution of the numerical value of \(z\) to get the values of \(x\) and \( y\). $$\begin{aligned}y &= - \frac{z}{2} \\ y &= - \frac{0}{2} \\ y &= 0\end{aligned}$$$$\begin{aligned}x &= y - z \\ x &= 0 - 0 \\ x &= 0\end{aligned}$$