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Percentages are a comparison of a number to 100. They're used often to describe increases, decreases, and various proportions. For instance, when we say something is 15% more, it means that it is 15 parts more out of 100 compared to the original value. In our problem, we are given that the woman earns 15% more than her husband. This can be translated mathematically as: if the husband's salary is represented by \(x\), then the wife's salary is \(x + 0.15x\), or \(1.15x\). This concept of 'percent increase' involves adding to the original amount. Understanding percentages is crucial in many life scenarios like calculating discounts, interest rates, etc. To convert a percentage to a decimal, simply divide by 100. For example, 15% becomes 0.15.

Linear equations form the backbone of algebra and involve expressions set equal to a value. They represent relationships where quantities change at a constant rate. In our problem, the combined income of the husband and wife is expressed as a linear equation: \(x + 1.15x = 69,875\). Here, \(x\) is a variable that we need to solve for, representing the husband's salary.

• Variables: Used as placeholders for the unknown value, in this case, \(x\) for the husband's salary.
• Constants: The numbers that are added or multiplied, like 1.15, which accounts for the 15% increase.
• Equation: Represents the sum of both incomes set equal to $69,875.

These equations are called 'linear' because they graph as a straight line and typically include no exponents. Solving them involves isolating the variable (\(x\) in our example) using basic arithmetic operations.

Problem solving involves several steps that help tackle mathematical and real-life problems efficiently. The first step is understanding the problem at hand—here, we needed to determine the husband's salary given certain conditions.

• Define the Unknown: Identify what you're trying to find (husband's salary) and assign a variable.
• Translate Words to Math: Convert the problem statement into a mathematical equation. In this case, we found out how much more the wife earns and expressed it in terms of the husband's earnings.
• Solve the Equation: Combine and simplify terms, then use arithmetic to solve for the unknown.
• Check Your Work: Re-evaluate the steps to ensure accuracy, making sure the solution fits all parts of the problem.

These systematic steps not only solve problems but build critical thinking skills useful in various fields. Problem solving is a methodical way of thinking that helps in resolving complex scenarios efficiently.