How to solve a linear equation with two unknowns when it is not a square matrix

Question

I am given the following two equations and am asked to solve for $x_1$ and $x_2$ using $A^{-1}$ (inverse method) of a matrix.

$$x_1 + 2x_2 = 1$$

$$3x_1 + 4x_2 = 5$$

how would you solve something like this? I thought to be able to do the inverse you would have to have a square matrix, and wouldn't this be a $2\times 3$ matrix?

Answer 1

Hint:

you have the equation:

$$[x_1,x_2]\left[\begin{eqnarray*} 1,3\\2,4 \end{eqnarray*}\right]=[1,5]$$

Answer 2

Your system in matrix form is

$\left[ \begin{array}{c} 1 \\ 5 \end{array} \right] = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right]$

Let $A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, B=\left[\begin{array}{c} 1 \\ 5 \end{array} \right]$ and $X=\left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right]$. Then you have $AX=B$. Now $A$ is a square matrix which is invertible. $A^{-1}=\begin{bmatrix} -2 & 1 \\ 3/2 & -1/2 \end{bmatrix}$ and $X=A^{-1}B$. $X$ is the solution matrix you need.