Problem 28 - College Algebra with Modeling and Visualization

Question

Write a system of linear equations with two variables whose solution satisfies the problem. State what each variable represents. Then solve the system. The sum of two numbers is 300 and their difference is \(8 .\) Find the two numbers.

Step-by-Step Solution

Answer

The numbers are 154 and 146.

Step 1: Define Variables

Let's represent the two numbers we need to find as variables. Let \( x \) be the larger number and \( y \) be the smaller number.

Step 2: Write Equations Based on the Problem Statement

We know from the problem:1. The sum of the two numbers is 300, so we can write the equation: \( x + y = 300 \).2. The difference between the two numbers is 8, which gives us the equation: \( x - y = 8 \).

Step 3: Solve the System of Equations Using Addition

To eliminate \( y \), add the two equations together:\[(x + y) + (x - y) = 300 + 8\]This simplifies to:\[2x = 308\]Divide both sides by 2 to find \( x \):\[x = 154\]

Step 4: Solve for the Second Variable

Substitute \( x = 154 \) back into the first equation:\[154 + y = 300\]Subtract 154 from both sides to solve for \( y \):\[y = 146\]

Step 5: Verify the Solution

Verify the solution by checking both conditions provided:1. The sum: \( 154 + 146 = 300 \).2. The difference: \( 154 - 146 = 8 \).Both conditions are satisfied, confirming that the solution is correct.

Key Concepts

Variables Definition

When solving a system of linear equations, the first step is defining the variables. This means we choose symbols, often letters like \(x\) and \(y\), to represent the unknown quantities in the problem. For our exercise, we need to find two numbers: one larger than the other. Therefore, we can define these variables as follows:

- Let \(x\) be the larger number.
- Let \(y\) be the smaller number.

Defining variables helps organize the solution. It acts as a bridge between the real-world problem and the mathematical model. This clear definition allows us to translate the word problem into equations that can be solved mathematically.

Solving Equations

Once variables are defined, we translate the problem into mathematical equations. The given conditions become the equations in our system. Here, the two conditions involve the sum and difference of two numbers. This gives us the following system of equations:

- The sum of the numbers: \( x + y = 300 \)
- The difference of the numbers: \( x - y = 8 \)

Next, we solve this system using the method of addition (or elimination). This method involves aligning the equations and adding them together to eliminate one of the variables. We'll add the equations to eliminate \( y \):

\[(x + y) + (x - y) = 300 + 8\]
This simplifies to \( 2x = 308 \). Dividing both sides by 2 gives \( x = 154 \).

Then, substitute \( x = 154 \) back into the first equation \( x + y = 300 \) to find \( y \):
\[154 + y = 300\]
Simplifying, we find \( y = 146 \).

This method provides a straightforward path to finding solutions by focusing on one variable at a time.

Verification of Solution

Verification is an essential step after solving equations. It ensures that the solution meets the original problem's conditions. To verify, substitute the found values back into the original equations.

For our problem, check the following conditions:

1. **Sum Check**: Verify that the sum of \( x \) and \( y \) is 300:
\[154 + 146 = 300\]
This condition is satisfied.

2. **Difference Check**: Verify that the difference between \( x \) and \( y \) is 8:
\[154 - 146 = 8\]
This condition is also satisfied.

When both conditions are met, it confirms that the solution is correct and the values we obtained for \( x \) and \( y \) solve the system of equations. Verification builds confidence in the solution and ensures that no errors were made in the process.