RWM102: Algebra

Example 1

A baker sells plain cakes for $7 and decorated cakes for $11. On a busy Saturday the baker started with 120 cakes, and sold all but three. His takings for the day were $991. How many plain cakes did he sell that day, and how many were decorated before they were sold?

The system of equations that describes this problem is:

$\begin{align*}p+d=117\!\\ 7p+11d=991\end{align*}$

Let's solve this system by substituting the second equation into the first equation:

$\begin{align*}p+d=117 \Rightarrow p=117-d\end{align*}$

$\begin{align*}7p+11d=991 & \Rightarrow 7(117-d)+11d=991\\ & \Rightarrow 819-7d+11d=991\\ & \Rightarrow 819+4d=991 \\ & \Rightarrow 4d=172 \\ & \Rightarrow d=43\end{align*}$

We can substitute $\begin{align*}d\end{align*}$ into the first equation to solve for $\begin{align*}p\end{align*}$.

$\begin{align*} p=117-d=117-(43)=74\end{align*}$

The baker sold 74 plain cakes and 43 decorated cakes.