Skill: Interpret solutions to systems in context

A solution to a system in context is an ordered pair that tells what both quantities are. The pair must satisfy both equations and make sense in the situation.

Example

At a school concert, adult tickets cost $9 and student tickets cost $6. A total of 8 tickets are sold for $54. How many of each type of ticket were sold?

1. Define the variables

Let

• $a$ = number of adult tickets
• $s$ = number of student tickets

The ordered pair will be $(a,s)$, so its first coordinate represents adult tickets and its second coordinate represents student tickets.

2. Write a system

The total number of tickets is 8:

$$a+s=8$$

The total cost is $54:

$$9a+6s=54$$

So the system is

$$\begin{cases} a+s=8\\ 9a+6s=54 \end{cases}$$

3. Solve the system

From the first equation,

$$s=8-a$$

Substitute this into the second equation:

$$9a+6(8-a)=54$$

Simplify:

$$9a+48-6a=54$$ $$3a=6$$ $$a=2$$

Now find $s$:

$$s=8-2=6$$

The solution is

$$(a,s)=(2,6)$$

4. Interpret the solution

Because $a$ represents adult tickets and $s$ represents student tickets, $(2,6)$ means:

2 adult tickets and 6 student tickets were sold.

Check both conditions:

• $2+6=8$ tickets
• $9(2)+6(6)=18+36=54$ dollars

The solution is meaningful because the quantities are nonnegative whole numbers.

If a system has no solution, the conditions are incompatible, so no combination works. If it has infinitely many solutions, the two conditions describe the same relationship, so many combinations work. In a real situation, restrictions such as nonnegative or whole-number quantities still determine which solutions make sense.