Chapter 4, Problem 16QC

To determine

To calculate: The system of equations that represent the number of registrations and T-shirts bought each year.

Answer to Problem 16QC

The system of equations is $\{\begin{array}{c}15x+10y=180\\ 12x+9y=150\end{array}$ . The number of registrations are 8 and T-shirts are 6.

Explanation of Solution

Given information:

The charge of 1 registration was $15 and one T-shirt was $10 so the student council paid $180 last year. Now, they paid $150 and charge of 1 registrations is $12 and cost of 1 T-shirt is $9.

Calculation:

Consider the provided information that charge of 1 registration was $15 and one T-shirt was $10 so the student council paid $180 last year. Now, they paid $150 and charge of 1 registrations is $12 and cost of 1 T-shirt is $9.

Let x denote the number of registrations and y denote the number of T-shirts.

Consider the system of equations that represent the situation above.

  $\begin{array}{c}15x+10y=180\\ 12x+9y=150\end{array}$

Use the method of substitution to solve the system above.

From the equation $12x+9y=150$ isolate the value of y ,

  $\begin{array}{c}12x+9y=150\\ y=\frac{150-12x}{9}\\ y=\frac{50-4x}{3}\end{array}$

Now substitute $y=\frac{50-4x}{3}$ in the equation $15x+10y=180$ ,

  $\begin{array}{c}15x+10\left(\frac{50-4x}{3}\right)=180\\ 45x+500-40x=540\\ 5x=40\\ x=8\end{array}$

Now, substitute $x=8$ in the equation $y=\frac{50-4x}{3}$ ,

  $\begin{array}{c}y=\frac{50-4\left(8\right)}{3}\\ y=6\end{array}$

Therefore, solution is the coordinate pair, $\left(8,6\right)$ .

Thus, the system of equations is $\{\begin{array}{c}15x+10y=180\\ 12x+9y=150\end{array}$ . The number of registrations are 8 and T-shirts are 6.