Question

Counting Calories Herschel uses an app on his smartphone to keep track of his daily calories from meals. One day his calories from breakfast were 125 more than his calories from lunch, and his calories from dinner were 300 less than twice his calories from lunch. If his total caloric intake from meals was \(2025,\) determine his calories for each meal.

Short answer

Lunch: 550 calories, Breakfast: 675 calories, Dinner: 800 calories.

Step-by-step solution

- Define variables

Let the calories from lunch be represented by the variable \( x \).

- Express other meals in terms of lunch

Calories from breakfast would be \( x + 125 \) and calories from dinner would be \( 2x - 300 \).

- Set up the equation

Add up the expressions for calories from all meals and set the sum equal to the total caloric intake: \( x + (x + 125) + (2x - 300) = 2025 \).

- Simplify the equation

Combine like terms: \( 4x - 175 = 2025 \).

- Solve for \( x \)

Add 175 to both sides: \( 4x = 2200 \). Then divide by 4: \( x = 550 \).

- Determine calories for each meal

Lunch: \( 550 \) calories. Breakfast: \( 550 + 125 = 675 \) calories. Dinner: \( 2(550) - 300 = 800 \) calories.

Key concepts

Linear Equations

Linear equations are an essential concept in algebra that you will encounter frequently. They are equations where the highest power of the variable is one. In our exercise, we use a linear equation to find out the number of calories Herschel consumed for his meals.

We start by creating an equation that represents the total calorie intake. The equation comes from adding up all the calories from breakfast, lunch, and dinner, which together equal 2025.

A linear equation looks like:
(x + (x + 125) + (2x - 300) = 2025). By solving this equation, we determine the calories for each meal. The steps include combining like terms, simplifying, and then isolating the variable to find its value. Linear equations help us connect different pieces of information and solve for unknown quantities efficiently.

Variables

Variables are symbols used to represent unknown values in mathematical expressions and equations. In this problem, we use the variable (x) to represent the calories from lunch.

By defining variables, we can simplify complex problems and solve them step-by-step. In our exercise, we define:

• Calories from lunch as (x).
• Calories from breakfast as (x + 125).
• Calories from dinner as (2x - 300).

.

With these variables, we can then create a single equation that represents the entire problem. Using variables is a powerful way to translate words and real-world situations into mathematical expressions, making them easier to work with and solve.

Caloric Intake

Caloric intake refers to the total number of calories a person consumes from food and drinks. It is important for managing weight and overall health. In this exercise, we track the calories Herschel consumes in a day from three meals: breakfast, lunch, and dinner.

Herschel's total caloric intake is given as 2025 calories. We use this information to set up an equation that includes the calories from all meals.

Knowing the distribution of calories across different meals helps in understanding eating patterns and nutritional balance. For example, in this exercise:

• Lunch calories are (550).
• Breakfast calories are (550 + 125 = 675).
• Dinner calories are (2(550) - 300 = 800).

By calculating the caloric intake accurately, we can ensure that our dietary needs are met and help maintain a healthy lifestyle. Tracking calories helps us make informed decisions about what and how much to eat.