Problem 80 - College Algebra

Question

The algebraic expression \(1527 x+31,290\) approximates average yearly earnings for elementary and secondary teachers in the United States \(x\) years after \(1990 .\) Evaluate the algebraic expression for \(x=10 .\) Describe what the answer means in practical terms.

Step-by-Step Solution

Answer

When \(x = 10\), the expression evaluates to $46,560. This represents the average yearly earnings for elementary and secondary teachers in the United States in the year 2000.

Step 1: Substitute the Value of \(x\)

Substitute \(x = 10\) into the algebraic expression \(1527x + 31290\). This gives \(1527 * 10 + 31290\).

Step 2: Simplify the Expression

Calculate the value of the expression. This means multiplying 1527 by 10 and then adding 31290 to the result. This results in \(15270 + 31290 = 46560\).

Step 3: Interpret the Result

The value we got, 46560, represents the average yearly earnings for elementary and secondary teachers in the United States 10 years after 1990, which is the year 2000. So, teachers earned approximately $46,560 in the year 2000 on average.

Key Concepts

Substitution Method

Substitution is a method used in algebra to replace variables with actual numbers to simplify computations or evaluate expressions. In this exercise, the expression provided is \(1527x + 31290\), which models the average yearly earnings of teachers. Here, \(x\) is the number of years after 1990. To find the earnings in the year 2000, we substitute \(x = 10\) because 2000 is 10 years after 1990. By substituting \(x = 10\) into the expression, we revise it to \(1527 \times 10 + 31290\). This is a crucial step in solving the equation because it allows us to use concrete numbers instead of abstract variables.

• Substitution helps simplify the process of finding specific values within an algebraic framework.
• It is particularly useful in transforming general models into specific predictions or outcomes.

Once the value is substituted, we can proceed to simplify and solve.

Simplification

Simplification refers to the process of reducing an expression to its simplest form, making it easier to understand and solve. After substituting \(x = 10\) in our expression \(1527x + 31290\), we need to carry out the arithmetic. The first step involves multiplying \(1527\) by \(10\), which results in \(15270\).
Next, we add this product to \(31290\), giving us \(46560\). Simply put, simplification condenses multiple steps and complex expressions into a more manageable form.

• It involves basic arithmetic operations such as addition, subtraction, multiplication, and division.
• Helps in clearly seeing the outcome of an equation or expression with less clutter.

By breaking down the problem step-by-step, simplification not only solves the equation but also reinforces the understanding of how the expression operates dynamically.

Interpretation of Results

Interpreting the results of a mathematical expression involves relating the simplified numeric outcome back to the real world context it models. In this case, after solving the equation, we end up with the number \(46560\). But what does \(46560\) mean?
This number represents the average yearly earnings of teachers 10 years after 1990, which calculates to the year 2000. Thus, in the year 2000, elementary and secondary teachers in the United States earned an average of $46,560. Interpreting results allows us to translate mathematical outcomes into information that offers insight or details applicable to real-world scenarios.

• It makes the abstract concepts of math tangible and understandable.
• Provides context that can inform decisions or further analysis.

Understanding what your solution represents is key in applying mathematical findings effectively.

Yearly Earnings Model

A yearly earnings model is a representation, often algebraic, used to predict or analyze income trends over a series of years. This exercise uses the model \(1527x + 31290\) to estimate teachers' earnings over time, where \(x\) denotes the number of years beyond 1990. Models like this are essential in economics and social sciences as they help in planning and policy formulation by offering insights into future trends and patterns.
Such models:

• Provide projections based on historical data.
• Help in budgetary estimates and financial forecasting.
• Offer a quantitative foundation for investigating economic questions.

Understanding how to utilize and interpret these models offers valuable skills in both academic and practical environments, highlighting the significance of algebraic expressions in real-world applications.