Problem 122 - Elementary Algebra

Question

When Gloria spent 15 minutes on the elliptical trainer and then did circuit training for 30 minutes, her fitness app says she burned 435 calories. When she spent 30 minutes on the elliptical trainer and 40 minutes circuit training she burned 690 calories. Solve the system \(\left\\{\begin{array}{l}15 e+30 c=435 \\ 30 e+40 c=690\end{array}\right.\) for \(e,\) the number of calories she burns for each minute on the elliptical trainer, and \(c,\) the number of calories she burns for each minute of circuit training.

Step-by-Step Solution

Answer

The number of calories Gloria burns per minute on the elliptical trainer is 4.2 calories, and per minute of circuit training is 8.4 calories.

Step 1: Set Up the Equations

Use the information provided to set up two equations. Let \( e \) represent the calories burned per minute on the elliptical, and \( c \) represent the calories burned per minute during circuit training. The equations based on the given data are: 1) \( 15e + 30c = 435 \) 2) \( 30e + 40c = 690 \)

Step 2: Simplify Equation 2

Divide the second equation by 10 to simplify it: \( 3e + 4c = 69 \)

Step 3: Multiply Equation 1 to Eliminate Terms

To eliminate \( e \), multiply the entire first equation by 2. This gives us: \( 2 * (15e + 30c) = 2 * 435 \) So, \( 30e + 60c = 870 \)

Step 4: Subtract the Simplified Equations

Subtract the equation from Step 2 (\( 3e + 4c = 69 \)) from the equation in Step 3 (\( 30e + 60c = 870 \)): \[ (30e + 60c) - (3e + 4c) = 870 - 69 \] Which simplifies to: \[ 27e + 56c = 801 \]

Step 5: Solve for \( c \)

With the simplified equation from Step 4, we can isolate \( c \). We subtract 27e from both sides: \[ 56c = 801 - 27e \] Solving for \( c \): \[ c = \frac{801 - 27e}{56} \]

Step 6: Substitute c back into Equation 1

Substitute \( c \) into the first equation \( 15e + 30c = 435 \). This gives: \[ 15e + 30 \times \frac{801 - 27e}{56} = 435 \]

Step 7: Solve for \( e \)

Multiply through to clear the fraction and solve for \( e \): \[ 15e + \frac{30 \times 801}{56} - \frac{30 \times 27e}{56} = 435 \ 15e + 4290 - \frac{810e}{56} = 435 \ \] Simplify and solve for \( e \): \[ \frac{450e}{56} = 435 - 4290 \ \frac{450e}{56} = -3855 \ e = \frac{-3855 \times 56}{450} \ e = 4.2 \ e = 4.2 \]

Step8: Solve for \( c \)

Using the value of \( e \) in \( 15e + 30c = 435 \), \[ 15(4.2) + 30c = 435 \ 63 + 30c = 435 \ 30c = 372 \ c = 12.4 \ c = 12.4 \]

Key Concepts

Calorie Burning

Calorie burning is a crucial factor in fitness and weight management. It refers to the number of calories you expend during physical activities. When you exercise, your body needs more energy, which it gets by burning calories.
Different exercises burn different amounts of calories, and the rate at which you burn them can be influenced by several factors, including:

• Intensity of the workout
• Duration of the exercise
• Your weight and metabolism
• The type of exercise

For instance, high-intensity workouts like circuit training can burn calories faster, while lower intensity exercises like walking might take longer to yield the same calorie burn.

Elliptical Trainer

An elliptical trainer is a popular piece of cardio equipment in gyms and homes. It's designed to simulate walking, running, or climbing stairs without causing excessive pressure to the joints. This makes it an ideal choice for people with joint issues or those wanting a low-impact workout.
Key benefits of using an elliptical trainer include:

• Effective calorie burning at a steady rate
• Low impact on joints, reducing injury risks
• Dual workout for both upper and lower body
• Customizable resistance levels

In the exercise problem, it's assumed that Gloria burns calories at a constant rate while using the elliptical trainer.

Circuit Training

Circuit training is a high-intensity workout that combines aerobic and resistance training. It involves performing a sequence of exercises, known as a 'circuit,' where each exercise targets different muscle groups. Here are some of the main benefits:

• Increased calorie burning
• Improved cardiovascular fitness
• Enhanced muscular strength and endurance
• Reduced workout time with high efficiency

In Gloria's case, circuit training adds to her overall calorie expenditure, and it's quantified per minute just like her time on the elliptical trainer.

Solving Systems of Linear Equations

In mathematics, solving a system of linear equations means finding values for variables that satisfy all equations in the system. Here's a brief on how it works:
Step-by-Step Solution of the Problem:
1. **Set Up the Equations:** Use the given data to create equations matching the problem's context, like Gloria’s workout times.
2. **Simplify the Equations:** Simplify if needed, often by dividing or multiplying through by constants.
3. **Eliminate Variables:** Use techniques like substitution or elimination to isolate one variable, making it easier to solve.
4. **Solve for One Variable:** Isolate and solve for one of the variables.
5. **Substitute Back:** Plug the solved value back into one of the original equations to find the second variable.
This method ensures a systematic approach to solving for unknown variables, as demonstrated in the exercise.