02 - Linear System - Example 2 - Solving Linear Systems (Infinite Solutions)

Determine the solutions (if any) to the following equation

$$x_1-x_2+x_3+3x_4=2;2x_1-x_2+x_3+2x_4=4;4x_1-3x_2+3x_3+8x_4=8$$

Solution (EROs)

There be less detail shown for the exact EROs but it is identical in concept to Linear System - Example 1

Trying to row-reduce the Augmented Matrix as much as possible

$$\left(\begin{array}{ccccc}1& -1& 1& 3& 2\\ 2& -1& 1& 2& 4\\ 4& -3& 3& 8& 8\end{array}\right)\overrightarrow{\begin{array}{rl}& A_{12}(-2)\\ & A_{13}(-4)\end{array}}\left(\begin{array}{ccccc}1& -1& 1& 3& 2\\ 0& 1& -1& -4& 0\\ 0& 1& -1& -4& 0\end{array}\right)\overrightarrow{\begin{array}{rl}& A_{21}(1)\\ & A_{23}(-1)\end{array}}\left(\begin{array}{ccccc}1& 0& 0& -1& 2\\ 0& 1& -1& -4& 0\\ 0& 0& 0& 0& 0\end{array}\right)$$

Converting back to the Linear System form

$$x_1-x_4=2;x_2-x_3-4x_4=0$$

As $x_3,x_4$ are loose variables (no restrictions on their value) then
Let

$$x_3=s,x_4=t$$

Leading us to

$$x_1=t+2,x_2=s+4t,x_3=s,x_4=t$$

Hence

$$(x_1,x_2,x_3,x_4)=(t+2,s+4t,s,t)=(2,0,0,0)+s(0,1,1,0)+t(1,4,0,1)$$

Geometric Intuition of the Solution ${\mathbb{R}}^4$ containing $(2,0,0,0)$ and parallel to $(0,1,1,0)$ and $(1,4,0,1)$

Plane in