Question

A buffet sells plates for seniors at $6, adults at $9, and children for free. If 150 plates were purchased for total receipts of $960 and twice as many adult plates were purchased as senior plates, how many of each type of plate were sold? 1)This problem can be solved using a system of equations. Identify the variables to be used in the system. 2)Write one equation to represent the total number of plates sold.3) Write one equation to represent the total receipts.4) Describe how a third equation can be written. What is that equation?5) Use a system of equations to solve the problem.

Answer 1

Question

1) Let take that $x$ plates for seniors, $y$ plates for adults and $z$ plates for children. Then our variables are: $x$, $y$ and $z$.

2) Total number of plates sold is $TN = x + y + z = 150$.

3) The total receipts: $TR = 6 \cdot x + 9 \cdot y + 0 \cdot z = 6 \cdot x + 9 \cdot y = 960$.

4) We know that twice as many adult plates were purchased as senior plates, so, the third equation can show us that the variable that represent the number of adult plates – variable $y$ – more by two times than the variable that represent the number of senior plates – variable $x$. So, we have: $y = 2 \cdot x$.

5) So, we have next system of equations:

$$\left\{ \begin{array}{l} x + y + z = 150 \\ 6 \cdot x + 9 \cdot y = 960 \Rightarrow \\ y = 2 \cdot x \end{array} \right. \Rightarrow \left\{ \begin{array}{l} x + 2 \cdot x + z = 150 \\ 6 \cdot x + 18 \cdot x = 960 \Rightarrow \\ y = 2 \cdot x \end{array} \right. \Rightarrow \left\{ \begin{array}{l} z = 150 - 3 \cdot x \\ 24 \cdot x = 960 \Rightarrow \\ y = 2 \cdot x \end{array} \right. \Rightarrow \left\{ \begin{array}{l} x = 150 - 3 \cdot 40 \\ x = 40 \\ y = 2 \cdot 40 \end{array} \right. \Rightarrow \left\{ \begin{array}{l} x = 40 \\ y = 80 \\ z = 30 \end{array} \right.$$

So, we find that: $x = 40, y = 80, z = 30$. Then we can say that 40 senior plates, 80 adult plates and 30 children plates were purchased.

**Answer**: Then we can say that 40 senior plates, 80 adult plates and 30 children plates were purchased.