Question

Solve the system by substitution:

$$\begin{gathered} 2x-y+z=-4 \\ z=5 \\ -2x+3y-z=-10 \end{gathered}$$

Answer 1

Step-1: Solve given equations:

Consider the given equations.

$$\begin{gathered} 2x-y+z=-4...\left(1\right) \\ z=5...\left(2\right) \\ -2x+3y-z=-10...\left(3\right) \end{gathered}$$

Solve the above equations by using the substitution method.

Substitute $z=5$ in equation $\left(1\right)$ and solve for $y$:

$\begin{array}{rcl}2x-y+5& =& -4\\ & \Rightarrow & y=2x+5+4\\ \therefore y& =& 2x+9...\left(4\right)\end{array}$

Step-2: Solve for $x$:

Substitute $y=2x+9$ and $z=5$ in equation $\left(3\right)$ and solve for $x$:

$\begin{array}{rcl}-2x+3\left(2x+9\right)-5& =& -10\\ & \Rightarrow & -2x+6x+27-5=-10\\ & \Rightarrow & 4x+22=-10\\ & \Rightarrow & 4x=-10-22\\ & \Rightarrow & 4x=-32\\ & \Rightarrow & x=\frac{-32}{4}\\ \therefore x& =& -8\end{array}$

Step-3: Solve for $y$:

Substitute $x=-8$ in the equation $\left(4\right)$ :

$\begin{array}{rcl}y& =& 2\left(-8\right)+9\\ & \Rightarrow & y=-16+9\\ \therefore y& =& -7\end{array}$

Hence, the value of $x$ is $\mathbf{-}\mathbf{8}$, $y$ is $\mathbf{-}\mathbf{7}$ and $z$ is $\mathbf{5}$.