Problem 61 - College Algebra

Question

Use a system of linear equations with two variables and two equations to solve. Find two numbers whose sum is 28 and difference is \(13 .\)

Step-by-Step Solution

Answer

The numbers are 20.5 and 7.5.

Step 1: Identify Variables

Let's identify the two unknown numbers as variables. Let \( x \) be the first number and \( y \) be the second number.

Step 2: Write Down Equations

We know two conditions: the sum of the numbers is 28 and their difference is 13. We can write these as two equations: \( x + y = 28 \) and \( x - y = 13 \).

Step 3: Solve the System of Equations

To solve the equations, we use the addition method. Add the two equations: \( (x + y) + (x - y) = 28 + 13 \). This simplifies to \( 2x = 41 \).

Step 4: Solve for the First Variable

Solve for \( x \) by dividing both sides of \( 2x = 41 \) by 2: \( x = \frac{41}{2} = 20.5 \).

Step 5: Solve for the Second Variable

Use the value of \( x \) in one of the original equations, for instance, \( x + y = 28 \). Substitute \( x = 20.5 \): \( 20.5 + y = 28 \). Solve for \( y \): \( y = 28 - 20.5 = 7.5 \).

Step 6: Verify the Solution

Verify by plugging \( x = 20.5 \) and \( y = 7.5 \) into the original equations. Ensure both conditions are met: \( 20.5 + 7.5 = 28 \) and \( 20.5 - 7.5 = 13 \). Both are true, hence the solution is correct.

Key Concepts

Variables

In solving systems of linear equations, the first step is to identify the variables. Variables act like placeholders for unknown values you need to find. In this problem, the two unknown numbers are what we're trying to discover, so we'll represent them as variables. Let's choose to call these variables \( x \) and \( y \).

• Let \( x \) be the first number.
• Let \( y \) be the second number.

By labeling our unknowns, we can easily set up equations to represent the problem mathematically. This makes it easier to solve for these missing values using algebraic methods.

Addition Method

The addition method is a powerful tool in solving systems of equations, particularly because it helps eliminate variables systematically. To use this method, manipulate the equations in such a way that adding them results in canceling one of the variables, simplifying the process greatly. Firstly, analyze your equations:

• \( x + y = 28 \)
• \( x - y = 13 \)

Add these equations:

• \((x + y) + (x - y) = 28 + 13\)
• This simplifies to \(2x = 41\)

Through addition, we've successfully eliminated the \( y \) variables, making it straightforward to solve for \( x \). This method is very helpful, especially when coefficients are chosen to cancel a variable without much extra manipulation.

Solving Equations

Once an equation with a single variable is isolated, solving becomes a matter of performing basic arithmetic operations. Using the obtained equation from the addition method, \( 2x = 41 \), you can solve for \( x \) directly:

• Divide both sides by 2: \( x = \frac{41}{2} = 20.5 \)

With \( x \) known, use it to find \( y \). Substitute \( x = 20.5 \) into one of the original equations, preferably the simpler one, such as \( x + y = 28 \):

• \( 20.5 + y = 28 \)
• Solve for \( y \): \( y = 28 - 20.5 = 7.5 \)

This step-by-step substitution ensures that both variables are found accurately, maintaining the balance and correctness of the original equations.

Verification

The final step in solving a system of equations is to verify your solutions. Verification is crucial to confirming that your answers are correct. Substitute both \( x \) and \( y \) back into the original equations to see if they hold true:

• \( x + y = 20.5 + 7.5 = 28 \)
• \( x - y = 20.5 - 7.5 = 13 \)

Both conditions are satisfied; thus, our solutions \( x = 20.5 \) and \( y = 7.5 \) check out. Verification is like a double-check, ensuring accuracy, and offering peace of mind that the solution is indeed the correct one for the given problem.