Q 4.32 - Intermediate Algebra

Question

Translate to a system of equations and then solve:

Erin spent $30$ minutes on the rowing machine and $20$ minutes lifting weights at the gym and burned $430$ calories. During her next visit to the gym she spent $50$ minutes on the rowing machine and $10$ minutes lifting weights and burned $600$ calories. How many calories did she burn for each minutes on the rowing machine? How many calories did she burn for each minute of weight lifting?

Step-by-Step Solution

Answer

The amount of calories burned per minute on the rowing machine and weight lifting are $11$ and $5$ respectively.

Step 1 . Given

We are given that Eric burns $430$ calories for spending $30$ minutes on rowing machine and $20$ minutes on lifting weights.

On the next visit she spent $50$ minutes on the rowing machine and $10$ minutes lifting weights and burned $600$ calories.

Step 2 . Explanation and Assumptions.

Let $x$ be the calories burnt per minute on the rowing machine and $y$ be calories burnt per minute on lifting weights.

The equations representing the calories burned are, 

$$\begin{gathered} 30x+20y=430 \\ 50x+10y=600 \end{gathered}$$

Step 3 . Solving the equations.

Multiplying by $2$ in the second equation, we get

$100x+20y=1200$

Now, subtracting the first equation from new equation, we get

$70x=770$

Dividing by $70$, we get

$$\begin{gathered} x=\frac{770}{70} \\ x=11 \end{gathered}$$

Step 4 . Finding y by putting the value of x .

Putting the value of $x$ in first equation, we get

$$\begin{gathered} 30\times 11+20y=430 \\ 330+20y=430 \\ 20y=430-330 \\ 20y=100 \end{gathered}$$

Dividing by $20$, we get

$$\begin{gathered} y=\frac{100}{20} \\ y=5 \end{gathered}$$

Step 4. Checking the solution

Checking the solution by putting the value of $x,y$ in the equations, we get

$$\begin{gathered} 30x+20y=430 \\ 30\left(11\right)+20\left(5\right)=430 \\ 330+100=430 \\ 430=430 \\ 50x+10y=600 \\ 50\left(11\right)+10\left(5\right)=600 \\ 550+50=600 \\ 600=600 \end{gathered}$$

This is true, hence the solution is correct.