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by **johnny23o** on Tue Aug 10, 2010 12:04 am

a movie theater charges $9.00 for adults and $7.00 for children. On a day when 325 people purchased tickets, the total receipts were $2675. How many adults tickets were sold? How many children tickets were sold? Create a system of linear equations and solve it using matrices.

I am really lost here please help

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by **Martingale** on Tue Aug 10, 2010 2:40 am

johnny23o wrote:a movie theater charges $9.00 for adults and $7.00 for children. On a day when 325 people purchased tickets, the total receipts were $2675. How many adults tickets were sold? How many children tickets were sold? Create a system of linear equations and solve it using matrices.

I am really lost here please help

Let X=#adult tickets
Let Y=#children tickets

now make two liner equations

by **johnny23o** on Tue Aug 10, 2010 1:20 pm

Let x= $9.00
let y= $7.00

9x+7y-325z=2675

would this work?

by **Martingale** on Tue Aug 10, 2010 1:55 pm

johnny23o wrote:Let x= $9.00
let y= $7.00

9x+7y-325z=2675

would this work?

no...you are letting x and y equal costs...let them equal the number of adult and children respectively. you should get two equations not one.

by **stapel_eliz** on Tue Aug 10, 2010 10:37 pm

johnny23o wrote:Let x= $9.00
let y= $7.00

9x+7y-325z=2675

would this work?

To learn the basic method for setting up this sort of exercise, try here. :wink:

by **Jaroslaw Jackiw** on Mon Sep 13, 2010 10:49 pm

Your problem states that $325$ people attended the theater that day, grossing $\$2675$ , at $\$9$ per ticket per adult ( $x_1$ )and $\$7$ per ticket per child ( $x_2$ ). This means that $\$9$ times $x_1$ , the number of adults, plus $\$7$ times $x_2$ , the number of children, produced $\$2675$ , and that the total number of adults, $x_1$ , plus children, $x_2$ , was $325$ . This is your system of linear equations:

$\begin{align} 9x_1 + 7x_2 &= 2675\\ x_1 + x_2 &= 325 \end{align}$

You mentioned that you needed to use matrices to solve the problem. Then you will need to convert your system of linear equations to matrix form, as follows:

$\begin{equation} \begin{bmatrix} 9 & 7\\ 1 & 1 \end{bmatrix} \begin{bmatrix} x_1\\ x_2 \end{bmatrix} = \begin{bmatrix} 2675\\ 325 \end{bmatrix} \end{equation}$

This will produce a matrix equation of the form $AX = B$ for:

$\begin{align} A &= \begin{bmatrix} 9 & 7\\ 1 & 1 \end{bmatrix}\\ X &= \begin{bmatrix} x_1\\ x_2 \end{bmatrix}\\ B &= \begin{bmatrix} 2675\\ 325 \end{bmatrix}\\ I &= \begin{bmatrix} 1 &0\\ 0 &1 \end{bmatrix} \end{align}$

such that:

$\begin{align*} AX &= B\\ A^{-1}AX &= A^{-1}B\\ IX &= A^{-1}B\\ X &= A^{-1}B \end{align*}$

Given $X$ and $B$ , you need to find $A^{-1}$ , the inverse matrix of $A$ , if it exists, to solve the equation. Using elementary row operations on $A^*$ , the augmented matrix of $A$ , allows you to obtain $A^{-1}$ :

$\begin{align*} A^* &= [A|I] \quad \rightarrow{} \quad \cdots \quad \rightarrow{} \quad [I|A^{-1}] = A^*_t\\ &= \begin{bmatrix} 9 &7 &| &1 &0\\ 1 &1 &| &0 &1 \end{bmatrix} \quad \rightarrow{} \quad \cdots \quad \rightarrow{} \quad \begin{bmatrix} 1 &0 &| &\frac{1}{2} &-\frac{7}{2}\\ 0 &1 &| &-\frac{1}{2} &\frac{9}{2} \end{bmatrix} \quad = A^*_t \end{align*}$

So that:

$\begin{align*} A^{-1} &= \begin{bmatrix} \frac{1}{2} &-\frac{7}{2}\\ -\frac{1}{2} &\frac{9}{2} \end{bmatrix}\\ &= \frac{1}{2} \begin{bmatrix} 1 &-7\\ -1 &9 \end{bmatrix} \end{align*}$

Since $X = A^{-1} B$ for:

$\begin{align} X &= \begin{bmatrix} x_1\\ x_2 \end{bmatrix}\\ A^{-1} &= \frac{1}{2} \begin{bmatrix} 1 &-7\\ -1 &9 \end{bmatrix}\\ B &= \begin{bmatrix} 2675\\ 325 \end{bmatrix} \end{align}$
then:

$\begin{align*} \begin{bmatrix} x_1\\ x_2 \end{bmatrix} &= \frac{1}{2} \begin{bmatrix} 1 &-7\\ -1 &9 \end{bmatrix} \begin{bmatrix} 2675\\ 325 \end{bmatrix}\\ &= \frac{1}{2} \begin{bmatrix} 2675 - 7 \cdot 325\\ 9 \cdot 325 - 2675 \end{bmatrix}\\ &= \frac{1}{2} \begin{bmatrix} 2675-2275\\ 2925-2675 \end{bmatrix}\\ &= \phantom{\frac{1}{2}} \begin{bmatrix} 200\\ 125 \end{bmatrix} \end{align*}$

Therefore, $x_1=200$ adults and $x_2=125$ children attended the theater that day.

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