Question

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
ANSWERED
asked 2021-02-09
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.)
\(\displaystyle{x}_{{1}}+{2}{x}_{{2}}+{6}{x}_{{3}}={6}\)
\(\displaystyle{x}_{{1}}+{x}_{{2}}+{3}{x}_{{3}}={3}\)
\(\displaystyle{\left({x}_{{1}},{x}_{{2}},{x}_{{3}}\right)}=\)?

Answers (1)

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:

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

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