Solve a system of equations using any method: word problems

Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Shawn and Abdul are hosting events that are catered by the same company. Shawn plans to have 64 adults and 93 children attend, so the total projected cost of his meals is $3,996. Abdul has 64 adults and 38 children on his guest list, so he will pay the caterer $3,336. How much does the caterer charge for each meal?

Every adult's meal costs $, and every child's meal costs $.

Answer 1

The information we are asked to find is the price of an adult meal, and the price of a child meal.

First, let's assign variables for those values, and adult meal costs $A and a child meal costs $C.

Looking at Shawn's plan, 64 adult meals will cost 64 times as much as 1 adult meal, and 93 child meals will cost 93 times as much as 1 child meal, so we can write an equation for his total cost as follows:

64A+93C=3996.

We can do the same for Abdul's plan.

64A+38C=3336

We notice that Shawn and Abdul are both planning for the same number of adults (64), so their total costs both include the cost of 64 adult meals. So, if we subtract Abdul's plan from Shawn's plan, we will have eliminated all the adult meals from our equation:

55C=660.

We now know that 55 child meals would cost $660.

By dividing the $660 by 55, we can find the cost of 1 child meal:

C=660/55=12.

We can now substitute this value into either Shawn's or Abdul's plans to find the cost of an adult meal:

Let's first try it with Shawn's plan:

64A+93(12)=3996

Simplifying:

64A+1116=3996

Subtracting 1116 from both sides:

64A=2880

And dividing:

A=2880/64=45

So an adult meal costs $45.

Now let's try it with Abdul's plan:

64A+38(12)=3336

Simplifying:

64A+456=3336

Subtracting 456 from both sides:

64A=2880

And dividing:

A=2880/64=45

So an adult meal costs $45.

Now we should restate our solutions: An adult meal costs $45, and a child meal costs $12.