Problem 174 - Elementary Algebra

Question

The total number of calories in 2 hot dogs and 3 cups of cottage cheese is 960 calories. The total number of calories in 5 hot dogs and 2 cups of cottage cheese is 1190 calories. How many calories are in a hot dog? How many calories are in a cup of cottage cheese?

Step-by-Step Solution

Answer

A hot dog has 150 calories. A cup of cottage cheese has 220 calories.

Step 1: Define Variables

Let the number of calories in a hot dog be represented by the variable \( h \), and the number of calories in a cup of cottage cheese be represented by the variable \( c \).

Step 2: Set Up Equations

From the given information, set up two equations:1. \( 2h + 3c = 960 \)2. \( 5h + 2c = 1190 \)

Step 3: Solve for One Variable

Solve the first equation for one variable in terms of the other. For instance, solve for \( h \):1. \( 2h + 3c = 960 \ 2h = 960 - 3c \ h = \frac{960 - 3c}{2} \)

Step 4: Substitute and Solve

Substitute \( h \) from the first equation into the second equation:1. \( 5 \left( \frac{960 - 3c}{2} \right) + 2c = 1190 \ 5 \left( 480 - 1.5c \right) + 2c = 1190 \ 2400 - 7.5c + 2c = 1190 \ 2400 - 5.5c = 1190 \ -5.5c = 1190 - 2400 \ -5.5c = -1210 \ c = \frac{-1210}{-5.5} \ c = 220 \)

Step 5: Find the Other Variable

Substitute \( c = 220 \) back into the equation for \( h \):1. \( h = \frac{960 - 3 \cdot 220}{2} \ h = \frac{960 - 660}{2} \ h = \frac{300}{2} \ h = 150 \)

Step 6: Verify the Solution

Verify the solution by substituting \( h = 150 \) and \( c = 220 \) into the original equations:1. \( 2(150) + 3(220) = 300 + 660 = 960 \) ✓2. \( 5(150) + 2(220) = 750 + 440 = 1190 \) ✓

Key Concepts

System of Linear Equations

A system of linear equations is a set of two or more linear equations containing the same variables. Each equation provides a constraint, and the solution to the system is the set of variable values that satisfies all the constraints simultaneously. In our exercise, we have two equations:

• Equation 1: 2h + 3c = 960
• Equation 2: 5h + 2c = 1190

The variables here are h (calories in a hot dog) and c (calories in a cup of cottage cheese). To solve this system, we need to find the values of h and c that satisfy both equations.

Substitution Method

The substitution method is a technique used to solve a system of linear equations. It involves solving one of the equations for one variable in terms of the other, and then substituting this expression into the other equation. This process effectively reduces the system to a single equation with one variable, making it easier to solve.

In our problem, we can solve the first equation for h:

• Equation 1: 2h + 3c = 960

Rearrange to solve for h:
2h = 960 - 3c
h = (960 - 3c) / 2

Next, we substitute this expression for h into the second equation:

• Equation 2: 5h + 2c = 1190

Substitute h:
5(480 - 1.5c) + 2c = 1190Now simplify and solve for c.

Variable Definition

Defining variables correctly is a critical step in solving any problem involving equations. Clear and appropriate definitions help ensure that the calculations are correct and that the results are meaningful.

In our exercise, we define the variables as follows:

• Let h represent the number of calories in one hot dog.
• Let c represent the number of calories in one cup of cottage cheese.

With these definitions, we can accurately set up our system of equations based on the problem statement.

Calorie Calculation

Once we have the variables and the system of linear equations set up, we can proceed with calculating the number of calories in each item. From solving our system, we found:

• The number of calories in one hot dog (h) is 150.
• The number of calories in one cup of cottage cheese (c) is 220.

These results were verified by substituting the values of h and c back into the original equations, and both equations held true:
2(150) + 3(220) = 960 ➔ True
5(150) + 2(220) = 1190 ➔ True

This confirms that our solution is correct, helping us accurately determine the calorie content of each food item.