Q. 362 - Intermediate Algebra

Question

In the following exercises,translate to a system of equations and solve.

Lynn paid a total of \(2,780 for 261 tickets to the theatre.student tickets cost \)10 and adult ticket cost is $15.How many Student tickets and how many adults tickets did Lynn buy?

Step-by-Step Solution

Answer

The number of adults tickets are 34 and the number of students tickets are 227.

Step 1.Given information.

Let x represent the number of adult tickets.

Let y represent the number of students' tickets.

The number of total tickets is 261.

That means,$x+y=261$.

Suppose the cost of an adult ticket is about $15, and the cost of a student ticket is about $10.

And the total cost of tickets at one performance is about $2,780.

That means,$15x+10y=2780$.

Step 2.Find the number of adult tickets and the number of student tickets were sold.

Consider the system of linear equations:$$\begin{gathered} x+y=261 \\ 15x+10y=2780 \end{gathered}$$

To solve the system of linear equations by substitution:

First, solve one equation for one variable:

Solve the first equation for x:

$$\begin{gathered} x+y=261 \\ y=261-x \end{gathered}$$

Step 3.Substitute y = 261 - x in the second equation 15 x + 10 y = 2780 and solve for x.

$$\begin{gathered} 15x+10y=2780 \\ 15x+10(261-x)=2780 \\ 15x+2610-10x=2780 \\ 5x+2610=2780 \\ 5x=2780-2610 \\ 5x=170 \\ x=34 \end{gathered}$$

Step 4.Substitute x = 34   i n t o   x + y = 261 and solve for y.

$$\begin{gathered} x+y=261 \\ 34+y=261 \\ y=261-34 \\ y=227 \end{gathered}$$

Step 5.Check

First substitute $x=34,y=277$ into the equation $x+y=261$.

$$\begin{gathered} x+y=261 \\ 34+227=261 \\ 261=261 \end{gathered}$$

This is true.

And also substitute $x=34,y=277$ into the equation $15x+10y=2780$.

$$\begin{gathered} 15x+10y=2780 \\ 15\left(34\right)+10\left(227\right)=2780 \\ 510+2270=2780 \\ 2780=2780 \end{gathered}$$

This is true.

Therefore, the number of adult tickets is 34 and the number of student tickets is 227.