Problem 134 - College Algebra Essentials

Question

It takes you 50 minutes to get to campus. You spend \(t\) minutes walking to the bus stop and the rest of the time riding the bus. Your walking rate is 0.06 mile per minute and the bus travels at a rate of 0.5 mile per minute. The total distance walking and traveling by bus is given by the algebraic expression $$ 0.06 t+0.5(50-t) $$ a. Simplify the algebraic expression. b. Use each form of the algebraic expression to determine the total distance that you travel if you spend 20 minutes walking to the bus stop.

Step-by-Step Solution

Answer

After simplification the algebraic expression becomes \(-0.44t + 25\). So, if you spend 20 minutes walking than you will travel a total of 16.2 miles.

Step 1: Simplify the algebraic expression

The first part of the problem requires simplifying the expression. We know that the expression is \(0.06t + 0.5(50 - t)\). To simplify it, we first apply the distributive property of multiplication over subtraction to \(0.5(50 - t)\). This gives us \(0.06t + 25 -0.5t\). In the next step, we consolidate the like terms which simplifies the expression to \(-0.44t + 25\).

Step 2: Substitute t=20 in the simplified expression

In the second part of the problem, we are asked to find the total distance travelled if we spend 20 minutes walking to the bus stop. This means we replace every 't' in our simplified expression \(-0.44t + 25\) with 20. Doing this gives us \(-0.44 * 20 + 25\).

Step 3: Solve the expression

Now we can solve the expression \(-0.44 * 20 + 25\). Giving us \(-8.8 + 25\) which equals to \(16.2\) miles.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a specific value or relationship. In our example, the expression \(0.06t + 0.5(50 - t)\) helps quantify the total distance travelled to the campus. The variables, like \(t\), represent unknowns and allow the expression to remain flexible depending on different input values. This expression incorporates both walking and bus travel rates to determine one combined result.

Understanding algebraic expressions is crucial because they form the foundation for solving equations and understanding relationships in math. They simplify complex real-world situations into manageable mathematical statements. Familiarizing oneself with the terminology related to algebraic expressions is a prime first step to mastering college algebra.

Simplifying Expressions

Simplifying expressions means rewriting them in a simpler or more concise form without changing their values. In the context of the original exercise, the expression \(0.06t + 0.5(50 - t)\) is simplifiable by expanding and combining like terms to make it easier to work with.This involves applying operations such as distributing multiplication across terms in parentheses and combining similar terms.

For instance, by distributing the \(0.5\) across \((50 - t)\), we break it down into \(0.5 * 50\) and \(-0.5 * t\), resulting in \(25 - 0.5t\). Then, combining it with the original \(0.06t\), we reduce the entire expression to \(-0.44t + 25\). Simplification makes evaluating the expressions faster and helps avoid errors in more complex algebraic computations.

Distributive Property

The distributive property is a foundational algebraic principle that states multiplication distributed over addition or subtraction is equal to a term multiplied by a group or each part of the group separately. In our exercise, this property is applied to \(0.5(50 - t)\), allowing us to distribute the multiplication separately to 50 and \(t\).

This transition changes the expression to \(0.5 * 50 - 0.5 * t\), leading to \(25 - 0.5t\). By using this property, complications in solving can often be simplified immediately, reducing chances for mistakes. It's useful for algebraic manipulations where dealing directly with complex groups or equations becomes cumbersome. Understanding the distributive property is essential in problem-solving across various math disciplines.

Substitution Method

The substitution method involves replacing a variable in an equation with a specific numerical value. This is a powerful technique used to evaluate expressions for given scenarios or initial conditions. In our example, to determine the total distance travelled when walking for 20 minutes, we substitute \(t\) in the simplified expression \(-0.44t + 25\) with 20.

This replacement gives us \(-0.44 * 20 + 25\), which calculates directly to 16.2 miles. Substitution helps turn abstract algebraic expressions into tangible real-world answers, providing clarity and full comprehension of the expression’s utility. These substitutions are integral, especially when variables have real-world implications, aiding in data-driven decision-making and precise solutions.