Problem 26 - Precalculus Mathematics for Calculus

Question

A woman earns 15% more than her husband. Together they make $69,875 per year. What is the husband’s annual salary?

Step-by-Step Solution

Answer

The husband's annual salary is $32,500.

Step 1: Define Variables

Let the husband's annual salary be \( x \). Since the woman earns 15% more than her husband, her salary is \( 1.15x \).

Step 2: Set Up the Equation

Together, their combined annual salary is $69,875. Therefore, we can write the equation: \[ x + 1.15x = 69875 \]

Step 3: Combine Like Terms

Combine the expressions for the salaries in the equation:\[ 2.15x = 69875 \]

Step 4: Solve for the Husband's Salary

To solve for \( x \), divide both sides of the equation by 2.15:\[ x = \frac{69875}{2.15} \]

Step 5: Calculate the Result

Perform the division to find the value of \( x \): \[ x = 32,500 \] Hence, the husband's annual salary is $32,500.

Key Concepts

Understanding Linear Equations

Linear equations are equations in which each term is either a constant or the product of a constant and a single variable. The equation used in this problem is a simple form of a linear equation. It's represented by the equation: \( x + 1.15x = 69875 \). This tells us something important: we're looking for a solution where the increase – represented by \( 1.15x \) – and the original amount \( x \), when combined, equal a total sum (in this case, \( 69875 \)).

Linear equations are foundational because they represent relationships and can easily show the change or difference, such as a percentage increase in salary. Remember, the heart of solving equations like these is to isolate the variable by understanding how each term contributes to the whole. Think of it as solving a puzzle where each piece plays a crucial role.

Combining Like Terms

When working with equations, especially like the one in this example, it’s important to combine like terms to simplify the problem. Like terms are terms whose variables and their exponents are the same. In this example, both terms \( x \) and \( 1.15x \) have the variable \( x \). To combine them:

• Add their coefficients: \( 1 + 1.15 = 2.15 \).
• This gives us \( 2.15x \).

If you combine like terms, the equation is much easier to work with because it reduces the number of terms, making the solving process straightforward and less prone to errors. In our case, simplifying \( x + 1.15x \) helps clarify how the increase affects the husband's salary, which simplifies our work into a single term.

Solving for a Variable

The ultimate goal in this problem is to solve for the variable \( x \), which represents the husband’s annual salary. To do so, we use the principle of isolation, which involves getting the variable by itself on one side of the equation with a coefficient of 1. Here's how it's done:

• We have the equation \( 2.15x = 69875 \).
• To isolate \( x \), divide both sides by 2.15: \( x = \frac{69875}{2.15} \).

By performing this division, we determine the exact amount of \( x \), which completes the solution. Solving for a variable is like peeling layers of an onion – you strip away the extra parts to reveal the crucial core. Here, the core is the husband's salary, which, once isolated, is calculated as $32,500. That's how solving equations helps us find unknown values in real-world contexts.