Originally Posted by
Matt Westwood Plug the values in.
11.0 = 4a + b
18.5 = 9a + b
Solve the simultaneous equations.
There are two ways to solve these equations. The simplest to understand is substitution. You can rearrange one of the equations (it doesn't matter which) so that $\displaystyle b$ is the subject*. Then substitute that expression into the second equation in place of $\displaystyle b$. Here's an example:
Suppose we have
$\displaystyle 10 = 5x + y$
$\displaystyle 12 = 7x + y$
We can take (for example) the first equation and write
$$\begin{aligned}
10 &= 5x + y \\
10 - 5x &= y \\
\end{aligned}$$
which is an expression for $\displaystyle y$. We then use that in the other equation
$$\begin{aligned}
12 &= 7x + y \\
12 &= 7x + (10 - 5x) \\
\end{aligned}$$
We can solve this equation for $\displaystyle x$, and when we have a value for $\displaystyle x$ we can put it into either of the original equations to find $\displaystyle y$.
$$\begin{aligned}
12 &= 7x + (10 - 5x) \\
2 &= 2x \\
x &= 1 \\[8pt]
10 &= 5x + y \\
10 &= 5(1) + y \\
10 &= 5 + y \\
5 &= y
\end{aligned}$$
The other approach is elimination. For this, we add (or subtract) multiples of the equations to make one of the variables disappear. In this case, since each equation has $\displaystyle 1y$, we can just subtract one equation from the other.
$$\begin{aligned}
12 &= 7x + y \\
10 &= 5x + y \\
12 - 10 &= (7x + y) - (5x + y) \\
2 &= 7x - 5x + y - y \\
2 &= 2x \\
x &= 1
\end{aligned}$$
And then we substitute our value for $\displaystyle x$ into one of the original equations as before.
* You can make $\displaystyle a$ the subject if you prefer, but it's a little more complicated.