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?

Short answer

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

Step-by-step solution

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.