Solving a $4\times 4$ linear system

Question

I was given the following problem and asked to solve for all the unknowns.

Solve for $x_1,x_2,x_3,x_4$: $$ \begin{matrix} x_1+x_2+x_3+x_4=1,\\ x_1+x_2-x_3=2,\\ x_2+x_3=0 \end{matrix} $$

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There are $4$ unknowns and $3$ equations. So isn't it not possible to get a single solution?

And every time I try to manipulate the equations, I keep on getting identity equations such as $$2=2$$

Answer 1

Hint

You're right. This system doesn't have a unique solution. Indeed it has infinitely many. Write the system, for example, as $$\left\{\begin{align}x_1+x_2+x_3& =1-x_4\\x_1+x_2-x_3 & =2\\x_2+x_3& =0\end{align}\right.$$ and solve it for $x_1,x_2,x_3.$ The solutions will depend on $x_4.$ For each value of $x_4$ you have a different solution.

Answer 2

you can take the solve variable as $$x_2,x_3,x_4$$ and the solution is given by $$\left\{\left\{x_2\to 1-\frac{x_1}{2},x_3\to \frac{x_1}{2}-1,x_4\to 1-x_1\right\}\right\}$$ and $x_1$ is a parameter.

Answer 3

By eliminating the third row, you have $$ \begin{cases}x_1+x_4=1 \\x_1+2x_2=2. \end{cases} $$ By simple observation, the solution can not be unique.

Now for the question regarding finding all the solutions to the system, do you know Gaussian elimination? What systematic method have you learned for solving linear equations?