# Solve the following system of equations. (Write your answers as a comma-separated list. If there are infinitely many solutions, write a parametric solution using t and or s. If there is no solution, write NONE.) x_1+2x_2+6x_3=6 x_1+x_2+3x_3=3 (x_1,x_2,x_3)=?

Equations
Solve the following system of equations. (Write your answers as a comma-separated list. If there are infinitely many solutions, write a parametric solution using t and or s. If there is no solution, write NONE.)
$$\displaystyle{x}_{{1}}+{2}{x}_{{2}}+{6}{x}_{{3}}={6}$$
$$\displaystyle{x}_{{1}}+{x}_{{2}}+{3}{x}_{{3}}={3}$$
$$\displaystyle{\left({x}_{{1}},{x}_{{2}},{x}_{{3}}\right)}=$$?

2021-02-10

Step 1
Given to solve the system of equations.
The system of equations can be solved using elimination.
To eliminate $$x_{­3}$$, the second equation is multiplied by 2 and subtracted from the first equation.

$$x_{1}+2x_{2}+6x_{3}=6$$

$$x_{1}+x_{2}+3x_{3}=3$$

$$(x_{1}+2x_{2}+6x_{3})-2(x_{1}+x_{2}+3x_{3})=6-2(3)$$

$$x_{1}+2x_{2}+6x_{3}-2x_{1}-2x_{2}-6x_{3}=6-6-x_{1}=0$$

$$x_{1}=0$$

Step 2

Plugging the value of $$x_{1}$$ in the first and second equations:

It is seen that the equations after plugging the value of $$x_{1}$$ are same. Hence, there are infinitely many solutions that satisfy the equation

$$x_{2} + 3x_{3} = 3$$

Hence, let $$x_{3} = t.$$

So the value of $$x_{2}$$ is given by:

Hence, the solution to the system of equations is given by

$$(x_{1}, x_{2}, x_{3})= (0, 3 – 3t, t)$$

$$(0)+2x_{2}+6x_{3}=6 \Rightarrow 2x_{2}+6x_{3}=6 \Rightarrow x_{2}+3x_{3}=3$$

$$(0)+x_{2}+3x_{3}=3 \Rightarrow x_{2}+3x_{3}=3$$

$$x_{2}+3x_{3}=3$$

$$x_{2}+3(t)=3$$

$$x_{2}+3t=3$$

$$x_{2}=3-3t$$

Step 3

Result:

$$(x_{1}, x_{2}, x_{3}) = (0, 3 – 3t, t)$$