Problem 131 - College Algebra

Question

will help you prepare for the material covered in the next section. Jane's salary exceeds Jim's by \(\$ 150\) per week. If \(x\) represents Jim's weekly salary, write an algebraic expression that models Jane's weekly salary.

Step-by-Step Solution

Answer

Jane's weekly salary can be represented by the algebraic expression \(x + 150\).

Step 1: Understanding the expression

Jane's salary exceeds Jim's by $150. This is the information given in the problem. If $x$ is Jim's weekly salary, Jane's weekly salary is Jim's weekly salary plus $150. This can be captured in a simple algebraic expression.

Step 2: Formulate the algebraic expression

According to the problem, Jane's salary is $150 more than Jim's salary. Therefore, if Jim's salary is represented by \(x\), Jane's salary can be represented as \(x + 150\) to reflect that Jane earns $150 more.

Key Concepts

Understanding Variables in Algebra

Variables are symbols used to represent unknown values or quantities in algebra. In this example, we use the variable \(x\) to represent Jim's weekly salary. Variables can take different forms, like letters or other symbols, and they allow us to write expressions without using specific numbers.
In our problem, the variable \(x\) stands for Jim's salary, which we don't have a specific number for. When you see \(x\) in an equation or expression, remember it's a placeholder for a number we either need to find or use for calculations.
This use of variables is crucial because it helps us model the situation with flexibility so we can figure out more complex problems. Variables are useful because they let us write general rules and expressions that can be adapted based on what we know.

Addition in Algebra

Addition in algebra works just like regular addition in math, but instead of adding just numbers, we can add variables, constants, or both.
In our problem with Jane and Jim's salaries, we're using addition to express the relationship between their earnings. Since Jane earns \(150 more than Jim, we add 150 to Jim's salary. This gives us the algebraic expression: \(x + 150\).
Here, \(x\) represents Jim's salary, and 150 is a constant that we add to it to find Jane's salary. So, if Jim's salary were, for instance, \)500, Jane's salary would be \(500 + 150\), which equals $650.
Addition in algebra helps us to see relationships and solve equations where values are connected by sums or differences.

Solving Word Problems with Algebraic Expressions

Word problems require converting real-world situations into algebraic expressions to solve them effectively. For Jane's and Jim's salaries, we need to interpret the context given in the words into mathematical terms.
When faced with a word problem like this, it's important to:

• Identify what you need to find—in this case, Jane's salary.
• Recognize what information you already have—Jim's salary and the additional $150 Jane earns.
• Use variables to represent unknown quantities—so, we assign \(x\) to Jim's weekly salary.

By transforming the problem's language into an algebraic expression \(x + 150\), we can easily manage the numbers and relationships involved.
This method of breaking down word problems into algebra helps us tackle real-life scenarios mathematically, ensuring clearer and more logical solutions.