Chapter 6.7, Problem 52STP

To determine

To find: Cost of each Activity

Answer to Problem 52STP

Hence the solution to the system of equation is $\left(5.75,4.25\right)$ . Therefore, amount of one game of miniature golf is $\$5.75$ and the amount of one go-karts ride is $\$4.25$ .

Explanation of Solution

Given:

T paid $\$25.75$ for $3$ games of miniature golf and $2$ rides on go-karts.

Trevor paid $\$35.75$ for $4$ games of miniature golf and $3$ rides on go-karts.

Calculation:

Suppose the amount of one game of miniature golf is $x$ and the amount of one ride on go-karts is $\$y$ .

T paid $\$25.75$ for $3$ games of miniature golf and $2$ rides on go-karts.

This gives $3x+2y=25.75$

Trevor paid $\$35.75$ for $4$ games of miniature golf and $3$ rides on go-karts.

This gives $4x+3y=35.75$

Therefore, system of equation is $\begin{array}{l}3x+2y=25.75\\ 4x+3y=35.75\end{array}$

Place the coefficients of the equations and the constant term into a matrix

  $\begin{array}{l}3x+2y=25.75\\ 4x+3y=35.75\end{array}$

  $\left[\begin{array}{l}\text{3 2 | 25}\text{.75}\\ \text{4 3 | 35}\text{.75}\end{array}\right]$

Therefore, augmented matrix is

  $\left[\begin{array}{l}\text{3 2 | 25}\text{.75}\\ \text{4 3 | 35}\text{.75}\end{array}\right]$

To make the first element in the first row a $1$ , multiply first row by $\frac{1}{3}$

  $\left[\begin{array}{l}\text{3 2 | 25}\text{.75}\\ \text{4 3 | 35}\text{.75}\end{array}\right]$

  $\left[\begin{array}{l}1\frac{2}{3}\text{| }\frac{25.75}{3}\\ \text{4 3 | 35}\text{.75}\end{array}\right]$

To make the first element in the second row a $0$ , multiply first row by $-4$ , and add the result to the second row

  $\left[\begin{array}{l}1\frac{2}{3}\text{ | }\frac{25.75}{3}\\ \text{4 3 | 35}\text{.75}\end{array}\right]$

  $\left[\begin{array}{l}1\frac{2}{3}\text{ | }\frac{25.75}{3}\\ \text{0 }\frac{1}{3}\text{ | }\frac{4.25}{3}\end{array}\right]$

To make the second element in the second row as $1$ , multiply second row by $3$

  $\left[\begin{array}{l}1\frac{2}{3}\text{ | }\frac{25.75}{3}\\ \text{0 }\frac{1}{3}\text{ | }\frac{4.25}{3}\end{array}\right]$

  $\left[\begin{array}{l}1\frac{2}{3}\text{ | }\frac{25.75}{3}\\ \text{0 1 | 4}\text{.25}\end{array}\right]$

To make the second element in the first row as $0$ , multiply second row by $\frac{-2}{3}$ and add the result to the first row

  $\left[\begin{array}{l}1\frac{1}{2}\text{ | }\frac{-1}{2}\\ \text{0 1 | }\frac{-5}{2}\end{array}\right]$

  $\left[\begin{array}{l}1\text{ 0 | }\frac{3}{4}\\ \text{0 1 | }\frac{-5}{2}\end{array}\right]$

Hence the solution to the system of equation is $\left(5.75,4.25\right)$ . Therefore, amount of one game of miniature golf is $\$5.75$ and the amount of one go-karts ride is $\$4.25$ .

Conclusion:

Hence, the solution to the system of equation is $\left(5.75,4.25\right)$ . Therefore, amount of one game of miniature golf is $\$5.75$ and the amount of one go-karts ride is $\$4.25$ .