Question

What is the solution of the system? Use elimination.

Answer 1

Answer:

The solutions to the system of the equations by the elimination method will be:

$x=2,\:z=-1,\:y=2$

Step-by-step explanation:

Given the system of the equations

$2x\:+\:2y\:+z\:=\:7$

$-x-\:y\:+z\:=\:-5$

$x+3y-4z=12$

solving the system of the equations by the elimination method

$\begin{bmatrix}2x+2y+z=7\\ -x-y+z=-5\\ x+3y-4z=12\end{bmatrix}$

$\mathrm{Multiply\:}-x-y+z=-5\mathrm{\:by\:}2\:\mathrm{:}\:\quad \:-2x-2y+2z=-10$

$\begin{bmatrix}2x+2y+z=7\\ -2x-2y+2z=-10\\ x+3y-4z=12\end{bmatrix}$

$-2x-2y+2z=-10$

$+$

$\underline{2x+2y+z=7}$

$3z=-3$

$\begin{bmatrix}2x+2y+z=7\\ 3z=-3\\ x+3y-4z=12\end{bmatrix}$

$2x+6y-8z=24$

$-$

$\underline{2x+2y+z=7}$

$4y-9z=17$

$\begin{bmatrix}2x+2y+z=7\\ 3z=-3\\ 4y-9z=17\end{bmatrix}$

Rearranging the equations

$\begin{bmatrix}2x+2y+z=7\\ 4y-9z=17\\ 3z=-3\end{bmatrix}$

solve $3z=-3$ for z:

$z=-1$

$\mathrm{For\:}4y-9z=17\mathrm{\:plug\:in\:}z=-1$

solve  $4y-9\left(-1\right)=17$ for y:

$4y-9\left(-1\right)=17$

$4y+9=17$

$4y=8$

$y=2$

$\mathrm{For\:}2x+2y+z=7\mathrm{\:plug\:in\:}z=-1,\:y=2$

solve $2x+2\cdot \:2-1=7$ for x:

$2x+2\cdot \:2-1=7$

$2x+3=7$

$2x=4$

$x=2$

Therefore, the solutions to the system of the equations by the elimination method will be:

$x=2,\:z=-1,\:y=2$