Example

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?

  plain cakes decorated cakes total
Cakes sold p d \(\begin{align*} 120-3=117\end{align*}\)
Cost of cakes 7p 11d $991

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.