Question

By using the method of matrices, solve the simultaneous equation.
2x - y = 4
3x - y = 5
Could someone explain how do this in detail with the working steps?

Answer 1

The first column will represent the coefficients of x, the second the coefficients of y, and the third will represent the constant on the right of each equation.

The first row represents the first equation, the second row represents the second equation.

The method used here is called Guass-Jordan elimination.\[\left[\begin{matrix}2 & -1 & 4\\ 3 & -1 & 5 \end{matrix}\right]{R_2-R_1\over \to}\left[\begin{matrix}2&-1&4\\1&0&1 \end{matrix}\right]\]The step above was to subtract row one from row two (R2-R1).

The next step is to switch row one and row two (not a necessary move, but helpful to me in this case):\[R_2\to R_1 \left[\begin{matrix}1 & 0&1 \\ 2 & -1&4\end{matrix}\right]{R_2-2R_1\over \to}\left[\begin{matrix}1 & 0&1 \\ 0 & -1&2\end{matrix}\right]\]The second move above was to subtract 2 times our new R2 from our new R1.

The final step is to multiply row two by -1:\[-R_2\left[\begin{matrix}1 & 0&1 \\ 0 & 1&-2\end{matrix}\right]\]Remembering what each row and column represents, we can see that this matrix is equivalent to the system\[x=1\]\[y=-2\]So we have our answer.

Note that this is just one of many ways to solve this system using matrices. Others include, but are not limited to, L-U decomposition and Cramer's rule.