Solving equations with two unknowns

In a nutshell

An equation with two unknowns is an equation with two variables that you do not know the value of. In some cases there may only be one solution, in others there may be a few and in many cases there could be infinite solutions!

Finding solutions to equations with two unknowns

Finding the solutions is all about using trial and error - trying out every possible solution.

Example 1

What are the possible whole number solutions to $2x + 4y = 16$ ?

$x$	$y$	$2x+4y$
$0$	$4$	$0+16=16$
$2$	$3$	$4+12=16$
$4$	$2$	$8+8=16$
$6$	$1$	$12+4=16$
$8$	$0$	$16+0 = 16$

There are five pairs of solutions: $\underline{(0,4) , (2,3) , (4,2) , (6,1)}$ and $\underline{(8,0)}$ .

Using a table

The easiest way to test out all the solutions is by using a table.

Example 2

Alex has some $20p$ and $50p$ coins. He has $£2.70$ all together. How many of each coin could he have?

Using a table:

$50p$	$20p$	$50x+20y=?$
$1$	$11$	$£2.70$
$2$	not possible
$3$	$6$	$£2.70$
$4$	not possible
$5$	$1$	$£2.70$

There are three possible combinations of coin that Alex could have:

$\underline{1\times50p}$ and $\underline{11 \times 20p}$

$\underline{3\times50p}$ and $\underline{6\times20p}$ 

$\underline{5\times50p}$ and $\underline{1\times20p}$ 