Q 92 - Intermediate Algebra

Question

The total amount of sodium in $2$  hot dogs and $3$ cups of cottage cheese is $4720mg$. The total
amount of sodium in $5$ hot dogs and $2$ cups of cottage cheese is $6300mg$. How much sodium is in a hot dog? How much sodium is in a
cup of cottage cheese?

Step-by-Step Solution

Answer

The total amount of sodium in hot dog is $860mg$ and the amount of sodium in a cup of cottage cheese is $1000mg$

Step 1. Given

The amount of sodium in $2$ hot dogs and $3$ cups of cottage cheese is $4720mg$

The amount of sodium in $5$ hot dogs and $2$ cups of cottage cheese is $6300mg$

Step 2. Form linear equations

Let $x$ denote the amount of sodium in hot dog.

And $y$denote the amount of sodium in cottage cheese.

The amount of sodium in $2$ hot dogs and $3$ cups of cottage cheese is $4720mg$

So, $2x+3y=4720$

The amount of sodium in $5$ hot dogs and $2$ cups of cottage cheese is $6300mg$

So, $5x+2y=6300$

Step 3. Solve the equation

Solve the system of linear equation

$2x+3y=4720$

$5x+2y=6300$

Solve the first equation for $x$

        $2x=4720-3y$

$x=\frac{4720-3y}{2}$

Step 4. Solve for y

Substitute $x=\frac{4720-3y}{2}$ in the second equation,

                   $5x+2y=6300$

$5\left(\frac{4720-3y}{2}\right)+2y=6300$

$\frac{23,600-15y+4y}{2}=6300$

          $23,600-11y=12,600$

                          $11y=11,000$

Divide both sides of the equation by $11$,

                              $y=1000$

The amount of sodium in cottage cheese is $1000mg$

Step 5. Solve for x

Substitute $y=1000$in the first equation,

          $2x+3y=4720$

$2x+3\left(1000\right)=4720$

                 $2x=4720-3000$

                 $2x=1720$

Divide both sides of the equation by $2$,

                   $x=860$

The amount of sodium in hot dog is $860mg$

Step 6. Check the answer

Substitute $x=860$,

$y=1000$ in the first equation,

                $2x+3y=4720$

$2\left(860\right)+3\left(1000\right)=4720$

       $1720+3000=4720$

                   $4720=4720$

Hence the equation holds true.