Q 90 - Intermediate Algebra

Question

Andrea is buying some new shirts and sweaters. She is able to buy $3$ shirts and $2$ sweaters for

$\backslash (114$ or she is able to buy $2$ shirts and $4$ sweaters for $\backslash )164$. How much does a shirt cost? How

much does a sweater cost?

Step-by-Step Solution

Answer

The cost of a shirt is $\$16$ and the cost of the sweater is $\$33$

Step 1. Given

The cost of $3$ shirts and $2$ sweaters is $\$114$

And the cost of $2$ shirts and $4$ sweaters is $\$164$

Step 2. Form linear equation

Let $x$ denote the cost of each shirt.

And $y$ denote the cost of each sweater.

The cost of $3$ shirts and $2$ sweater is $\$114$

So, $3x+2y=114$

The cost of $2$ shirts and $4$ sweaters is $\$164$

So, $2x+4y=164$

Step 3. Solve linear equation

Solve the linar equations,

$3x+2y=114$

$2x+4y=164$

Divide the second equation by $2$,

   $x+2y=82$

          $x=82-2y$

Step 4. Solve for y

Substitute $x=82-2y$ in the first equation,

             $3x+2y=114$

$3(82-2y)+2y=114$

  $246-6y+2y=114$

          $246-4y=114$

                    $4y=246-114$

                    $4y=132$

Divide both sides of the equation by $4$,

                      $y=33$

The cost of the sweater is $\$33$

Step 5. Solve for x

Substitute $y=33$ in the second equation,

        $2x+4y=164$

   $2x+4\left(33\right)=164$

                $2x=164-132$

                $2x=32$

Divide both sides of the equation by $2$,

                  $x=16$

The cost of the shirt is $\$16$

Step 6. Chack the answer

Substitute $x=16$,

                  $y=33$ in the first equation,

         $3x+2y=114$

$3\left(16\right)+2\left(33\right)=114$

         $48+66=114$

               $114=114$

Hence the equation holds true.