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What is the solution of the system? Use elimination.

2x + 2y +z = 7
-X – y +z = -5
x + 3y – 4z = 12

User Stojke
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1 Answer

2 votes

Answer:

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


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

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

User Afroz
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