Q 93 - Intermediate Algebra

Question

The total number of calories in $2$ hot dogs and $3$ cups of cottage cheese is $960$ calories. The total

number of calories in $5$ hot dogs and $2$ cups of cottage cheese is $1190$ calories. How many calories are in a hot dog? How many calories are in a cup of cottage cheese?

Step-by-Step Solution

Answer

The number of calories in hot dog is $150$ and the number of calories in $220$

Step 1. Given

The number of calories in $2$ hot dogs and $3$ cups of cottage cheese is $960$

The number of calories in $5$ hot dogs and $2$ cups of cottage cheese is $1190$

Step 2. Form the linear equation

Let  denote the number of calories in hot dog.

And denote the number of calories in cottage cheese.

The number of calories in $2$ hot dogs and $3$ cups of cottage cheese is $960$

So, $2x+3y=960$

The number of calories in $5$ hot dogs and $2$ cupsof cottage cheese is $1190$

So, $5x+2y=1190$

Step 3. Solve the linear equation

Solve the system of linear equations.

$2x+3y=960$

$5x+2y=1190$

Solve the first equation for $x$,

  $2x=960-3y$

   $x=\frac{960-3y}{2}$

Step 4. Solve for y

Substitute $x=\frac{960-3y}{2}$ in the second equation,

                   $5x+2y=1190$

  $5\left(\frac{960-3y}{2}\right)+2y=1190$

   $\frac{4800-15y+4y}{2}=1190$

              $4800-11y=2380$

                          $11y=4800-2380$

                         $11y=2420$

Divide both sides of the equation by $11$,

                              $y=220$

The number of calories in cottage cheese is $220$

Step 5. Solve for x

Substitute $y=220$ in the second equation,

         $5x+2y=1190$

  $5x+2\left(220\right)=1190$

                 $5x=1190-440$

                 $5x=750$

Divide both sides of the equation by $5$,

                   $x=150$

The number of calories in hot dog is  $150$ calories.

Step 6. Check the answer

Substitute $x=150$,

                  $y=220$ in the first equation,

              $2x+3y=960$

$2\left(250\right)+3\left(220\right)=960$

         $300+660=960$

                   $960=960$

Hence the equation holds true.