Best way to solve $2$ equations in an efficient manner?

Question

I will soon be taking an examination that will require me to solve $2$ simultaneous equations of the form $ax+by=0$ and $cx+dy=k$. I have access to a calculator but it does not solve these types of problems for me unfortunately.

The numbers are not nice to work with either like $a=0.479374, b=3.485493$ etc. And there will be a lot of time restrictions and pressure.

I wonder what is the best way to solve such a system of equations?

Answer 1

Solve for $x$ and/or $y$ in terms of $a,b,c,d,k$.

$$x = \frac{k}{c - \frac{db}{a}} = \frac{ka}{ca-db}$$

$$y = -\frac{b}{a}x$$

In this case, plug in $a,b,c,d,k$ to find the solution for $x$, and then use that value for $x$ to find $y$.

Answer 2

Plug and chug

$$ \begin{align} g & = a d - b c \\ x & =-b \frac{k}{g} \\ y & = a\frac{k}{g} \end{align} $$