Chapter 3.1, Problem 56PPSE

Solution

To explain how to solve a system of equations using tables to find the number of $4\times 6$ prints ordered.

Let $x$ denotes the number of $5\times 7$ prints and $y$ denotes the number of $4\times 6$ prints.

Model the given situation mathematically using a system of linear equations:

  $\left\{\begin{array}{c}y=6-x\\ y=30-7x\end{array}\right.$ .

Solve the system by using the method of tables:

  $x=4$ and $y=2$ .

Then, it follows that, the number of $4\times 6$ prints ordered is $2$ .

Given:

Given that Amy ordered a total of $6$ photographs for $\$7.50$ .

The cost of $5\times 7$ and $4\times 6$ prints are given to be $\$1.75$ and $\$0.25$ respectively.

Calculation:

Model the given situation mathematically using a system of linear equations:

Suppose that $x$ denotes the number of $5\times 7$ prints and $y$ denotes the number of $4\times 6$ prints.

Since Amy ordered a total of $6$ photographs, it follows that:

  $x+y=6$ .

Now, since the cost of a $5\times 7$ print is $\$1.75$ , the total cost of $5\times 7$ prints in dollars:

  $1.75x$ .

Likewise, the total cost of $4\times 6$ prints:

  $0.25y$ .

Then, the total cost of the order of photographs:

  $1.75x+0.25y$ .

The total cost of the order is given to be $\$7.50$ .

Then, it follows that:

  $1.75x+0.25y=7.50$ .

Then, a system of linear equations modelling the given situation is found:

  $\left\{\begin{array}{c}x+y=6\\ 1.75x+0.25y=7.50\end{array}\right.$ .

Solve the system of equations using tables:

Consider the system of equations $\left\{\begin{array}{c}x+y=6\\ 1.75x+0.25y=7.50\end{array}\right.$ .

Consider the first equation $x+y=6$ .

Isolate $y$ on the left side of the equation:

  $y=6-x$ .

Now, consider the second equation $1.75x+0.25y=7.50$ and do the same process to observe:

  $y=30-7x$ .

Then, the system of equation reduces to:

  $\left\{\begin{array}{c}y=6-x\\ y=30-7x\end{array}\right.$ .

Now, use the tabular approach to solve the system of equations by listing $x$ -values until the corresponding $y$ -values match:

$x$	$y=6-x$	$y=30-7x$
$1$	$6-1=5$	$30-7\cdot 1=23$
$2$	$6-2=4$	$30-7\cdot 2=16$
$3$	$3$	$9$
$4$	$2$	$2$

Observe that the $y$ -values match when $x=4$ .

Substitute $x=4$ in any of the equations, say $y=6-x$ , to obtain the value of $y$ :

  $\begin{array}{l}y=6-4\\ y=2\end{array}$ .

Thus, the value of $y$ is $2$ .

That is, the number of $4\times 6$ prints ordered is $2$ .

Conclusion:

Let $x$ denotes the number of $5\times 7$ prints and $y$ denotes the number of $4\times 6$ prints.

Model the given situation mathematically using a system of linear equations:

  $\left\{\begin{array}{c}y=6-x\\ y=30-7x\end{array}\right.$ .

Solve the system by using the method of tables:

  $x=4$ and $y=2$ .

Then, it follows that, the number of $4\times 6$ prints ordered is $2$ .