Problem 21 - Introductory Algebra for College Students

Question

Use the five-step strategy to solve each problem. How will you spend your average life expectancy of 78 years? The bar graph shows the average number of years you will devote to each of your most time-consuming activities. Exercises \(21-22\) are based on the data displayed by the graph. (GRAPH CANNOT COPY) According to the American Bureau of Labor Statistics, you will devote 37 years to sleeping and watching TV. The number of years sleeping will exceed the number of years watching TV by 19. Over your lifetime, how many years will you spend on each of these activities?

Step-by-Step Solution

Answer

You will spend 28 years sleeping and 9 years watching TV over your lifetime.

Step 1: Formulate the algebraic equations

Let's denote sleeping as \( S \) and watching TV as \( T \). Since the time devoted to these two activities total 37 years, you can formulate the first equation as \( S + T = 37 \). Next, the problem mentions that the time spent sleeping exceeds the time spent watching TV by 19 years. This forms the second equation which can be written as \( S = T + 19 \). Now you have two equations.

Step 2: Solve the simultaneous equations

There are different ways to solve these equations. One way is to use the second equation to substitute for \( S \) in the first equation. From \( S = T + 19 \), extract \( T \) to be \( S - 19 \). You can now replace \( T \) in the first equation, which becomes \( S + (S - 19) = 37 \). Solve this equation to get \( S = 28 \). With \( S \) now known, you can substitute it into the second equation to get \( T = 28 - 19 \), hence \( T = 9 \). These values represent the number of years you will spend on each of these activities over your lifetime.

Key Concepts

Understanding Simultaneous Equations

When tackling algebra problems, simultaneous equations are quite common. These equations involve two or more equations rated by two or more unknowns that require solving together. In the context of the original problem, the equations look like this:

• The total time dedicated to sleeping (\( S \)) and watching TV (\( T \)) is 37 years, which gives us our first equation, \( S + T = 37 \).
• The time spent sleeping exceeds the time spent watching TV by 19 years, leading to the second equation, \( S = T + 19 \).

These two equations need to be solved simultaneously. This technique is crucial in applications where two tasks or conditions must be met concurrently. By substituting and manipulating variables, we can uncover the exact values of the unknowns.

Delving into Algebraic Equations

An algebraic equation forms the backbone of many problem-solving processes. In algebra, equations are mathematical statements that assert the equality of two expressions. They are vital as they allow us to find unknown values through manipulation.
In our specific problem, algebraic equations were used to translate real-world activities into mathematical terms. Here are the steps used:

• First, we made a clear definition by assigning variables to unknown quantities: \( S \) for sleeping and \( T \) for watching TV.
• Next, we formulated equations using these variables based on relationships given in the problem. Without these equations, direct calculation is often impossible.

Once arranged as algebraic equations, finding solutions comes down to rules and operations, such as adding, subtracting, or substituting. This turns complex real-life situations into solvable equations.

Crafting a Problem-Solving Strategy

Tackling any algebra problem efficiently involves developing a robust problem-solving strategy. This strategy involves clear steps to understand and solve an issue effectively. Here are some key steps used in solving our original problem:

• Understand the problem: Grasp the essential elements by breaking down the problem statement to its key parts.
• Formulate equations: Transform the given information into algebraic equations. Knowing what each variable represents is crucial.
• Solve the equations: Use methods such as substitution, elimination, or matrix operations to deduce the unknowns.
• Verify and interpret: Check your solutions to ensure they satisfy all original conditions. Interpreting the answer in real-world terms is essential for clarity.

By methodically organizing thoughts and steps, problem-solving transforms from a daunting task into an achievable goal.