Write the following in the form of AX = B

Question

Write the following system of equations in the form $AX = B$, and calculate the solution using the equation $X = A^{-1}B$.

$$2x - 3y = - 1$$

$$-5x +5y = 20$$

I'm not the strongest at linear algebra but I don't understand what the question is asking me over here or how to even go about solving this.

Answer 1

Let $X = \left[\begin{matrix}x\\y\end{matrix}\right]$, then the first equation can be written, in matrix multiplication, as $$\left[\begin{matrix}2&-3\end{matrix}\right]\left[\begin{matrix}x\\y\end{matrix}\right] = \left[\begin{matrix}-1\end{matrix}\right]$$

Similarly, the second equation can be written as $$\left[\begin{matrix}-5&5\end{matrix}\right]\left[\begin{matrix}x\\y\end{matrix}\right] = \left[\begin{matrix}20\end{matrix}\right]$$

By combining these two equations as a system, they can be written as $$\begin{align*}\left[\begin{matrix}2&-3\\-5&5\end{matrix}\right]\left[\begin{matrix}x\\y\end{matrix}\right] &= \left[\begin{matrix}-1\\20\end{matrix}\right]\\ AX &= B \end{align*}$$

Now, if the above $A$ has an inverse, then both sides can be left-multiplied by $A^{-1}$ to get $$\begin{align*}AX &= B\\ A^{-1}AX &= A^{-1}B\\ I_2X &= A^{-1}B\\ X &= A^{-1}B\\ \left[\begin{matrix}x\\y\end{matrix}\right] &= \left[\begin{matrix}2&-3\\-5&5\end{matrix}\right]^{-1}\left[\begin{matrix}-1\\20\end{matrix}\right] \end{align*}$$

And then the original unknowns in the system of equation can be solved.

Answer 2

HINT: You have a set of linear equations.
These can be written in Matrix form: $$AX=B$$ So you can build A by using the coefficients of x and y: $$A=\begin{bmatrix} 2 & -3 \\ -5 & 5 \\ \end{bmatrix}$$

X is the unknown variables x and y and it is a Vector:
$$X=\begin{bmatrix} x\\ y\\ \end{bmatrix}$$

And the multiplication of Matrix A with vector X is the solution vector B: $$B=\begin{bmatrix} -1\\ 20\\ \end{bmatrix}$$

To solve for X in linear algebra you cannot divide directly, but accomplish this by taking the inverse of A and multiply it by B.