How to solve these equations?

Question

I have come across two equations while deriving some trigonometric identities. How to obtain the values of $x$ and $y$ when $a$,$b$,$c$ and $d$ are known values? $$ ax + by = c$$ $$ ay - xb = d$$ Please pardon me if the question seems trivial. Any suggestions will help.

NOTE: Thank you for various solutions. The obvious solution of elimination did not come in my mind. Sorry for this basic question.

Answer 1

Well you can use substitution to find $x$ and $y$.

$$ax + by =c $$ $$ y = \frac{c - ax}{b}$$

then you would substitute the $y$ equation to solve for $x$

$$ a(\frac{c - ax}{b}) - xb = d$$

Then once you solved for $x$, you can solve for $y$!

For example, let's say $a = 1, b = 2, c =3 , d= 4 $

Your two system of equations would be

$$ \tag{1} x + 2y = 3 $$ $$ \tag{2} y - 2x = 4 $$

Solving $(1)$ for $y$:

$$y = \frac{3-x}{2}$$ Then sub the $y$ equation into $(2)$, we have:

$$ \frac{3-x}{2} - 2x = 4 $$

then $$\color{red}{x = -1}$$

sub that back into $(1)$ and solve for $y$

$$ -1 + 2y = 3 $$ $$ \color{red}{y = 2} $$

I hope this help!

Answer 2

Guide:

If you have learn about matrices, the problem is equivalent to the following:

$$\begin{bmatrix} a & b \\ - b & a \end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} c \\ d \end{bmatrix}$$

Consider taking inverses of a particular matrix.