Q 91 - Intermediate Algebra

Question

Peter is buying office supplies. He is able to buy $3$ packages of paper and $4$ staplers for $\backslash (40$ or he

is able to buy $5$ packages of paper and $6$ staplers for $\backslash )62$. How much does a package of paper cost? How much does a stapler cost?

Step-by-Step Solution

Answer

The cost of package of paper is $\$4$ and the cost of stapler is $\$7$

Step 1. Given

The cost of $3$ package of paper and $4$ stapler is $\$40$.

The cost of $5$ paper package paper and $6$ staplers is $\$62$

Step 2. Form linear equations

Let $x$ denote the cost of package of paper and $y$ denote the cost of stapler.

The cost of $3$ paper package and $4$ stapler is $\$40$

So, $3x+4y=40$

The cost of $5$ paper package and $6$ stapler is $\$62$

So, $5x+6y=62$

Step 3. Solve the linear equation

Solve the system of linear equations.

$3x+4y=40$

$5x+6y=62$

Solve the first equation for $x$,

        $3x=40-4y$

          $x=\frac{40-4y}{3}$

Step 4. Solve for y

Substitute $x=\frac{40-4y}{3}$ in the second equation,

               $5x+6y=62$

$5\left(\frac{40-4y}{3}\right)+6y=62$

$\frac{200-20y}{3}+6y=62$

Take LCM and solve 

$200-20y+18y=186$

                      $2y=200-186$

                       $2y=14$

Divide both sides of the equation by $2$,

                         $y=7$

The cost of the stapler is $\$7$

Step 5. Solve for x

Substitute $y=7$ in the first equation,

        $3x+4y=40$

        $3x+28=40$

                $3x=40-28$ 

                $3x=12$

Divide both sides of the equation by $3$,

                  $x=4$

The cost of the paper package is $\$4$

Step 6. Check the answer

Substitute $x=4$,

                  $y=7$ in the first equation,

        $3x+4y=40$

    $3\left(4\right)+4\left(7\right)=40$

        $12+28=40$

                $40=40$

Hence the equation holds true.