Question

# Solve the following system of equations. 2x_1−x_2−x_3=−3 3x_1+2x_2+x_3=13 x_1+2x_2+2x_3=11 (x_1, x_2, x_3) =

Equations
Solve the following system of equations.
$$\displaystyle{2}{x}_{{1}}−{x}_{{2}}−{x}_{{3}}=−{3}$$
$$\displaystyle{3}{x}_{{1}}+{2}{x}_{{2}}+{x}_{{3}}={13}$$
$$\displaystyle{x}_{{1}}+{2}{x}_{{2}}+{2}{x}_{{3}}={11}$$
$$\displaystyle{\left({x}_{{1}},{x}_{{2}},{x}_{{3}}\right)}=$$

2021-01-14
Step 1
Let x1=x , x2=y and x3=z
So, the equations are:
2x-y-z=-3
3x+2y+z=13
x+2y+2z=11
Step 2
Let use eliminate z first. Add first two equations. That gives 5x+y=10
Step 3
Multiply first equation by 2 and add it to the third equation.
$$\displaystyle{2}{\left({2}{x}-{y}-{z}=-{3}\right)}\Rightarrow{4}{x}-{2}{y}-{2}{z}=-{6}$$
So we get 5x=5
So, $$\displaystyle{x}=\frac{{5}}{{5}}={1}$$
Step 4
Here we plug x=1 and find y
5(1)+y=10
Or, 5+y=10
Or, y=5
Step 5
In 2x-y-z=-3 we plug x=1 and y=5 and find z
2(1)-5-z=-3
Or, 2-5-z=-3
Or, -3-z=-3
Or, z=0
Result: $$\displaystyle{\left({x}_{{1}},{x}_{{2}},{x}_{{3}}\right)}={\left({1},{5},{0}\right)}$$