Question

Write a system of equations to describe the situation below, solve using an augmented matrix.The glee club needs to raise money for the spring trip to Europe, so the members are assembling holiday wreaths to sell. Before lunch, they assembled 20 regular wreaths and 16 deluxe wreaths, which used a total of 140 pinecones. After lunch, they assembled 20 regular wreaths and 18 deluxe wreaths, using a total of 150 pinecones. How many pinecones are they putting on each wreath?The regular wreaths each have ? pinecones on them and the large ones each have ? pinecones.

Answer 1

Given

First : They assembled 20 regular wreaths and 16 deluxe wreaths, which used a total of 140 pinecones

Let's represent regular wreaths with r

and

Let's represent deluxe wreaths with d

`20r+16d=140`

Second: They assembled 20 regular wreaths and 18 deluxe wreaths, using a total of 150 pinecones

`20r+18d=150`

We now have

`\begin{gathered} 20r+16d=140\text{ ...Equation 1} \\ 20r+18d=150\text{ ...Equation 2} \end{gathered}`

we can now solve simultaneously, by subtracting the equation 1 and 2

`\begin{gathered} 20r-20r+16d-18d=140-150 \\ -2d=-10 \\ Divide\text{ both sides by -2} \\ -(2d)/(-2)=-(10)/(-2) \\ \\ d=5 \end{gathered}`

We can subsitute d=5 in equation 1 or 2

`\begin{gathered} 20r+16d=140\text{ ... Equation 1} \\ 20r+16(5)=140 \\ 20r+80=140 \\ 20r=140-80 \\ 20r=60 \\ divide\text{ both sides by 20} \\ (20r)/(20)=(60)/(20) \\ \\ r=3 \end{gathered}`

The final answer

5 pinecones on the deluxe wreath and 3 pinecones on the regular wreath