Problem 22 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. In a recent school election, 571 votes were cast for class president. If the winner received 89 more votes than the loser, how many votes did each receive?

Step-by-Step Solution

Answer

The loser received 241 votes and the winner received 330 votes.

Step 1 - Understand the Problem

Read the problem carefully. Identify the total number of votes cast and the relationship between the winner's and the loser's votes.

Step 2 - Define the Variables

Let the number of votes that the loser received be \( x \). Since the winner received 89 more votes than the loser, the number of votes the winner received can be expressed as \( x + 89 \).

Step 3 - Set Up the Equation

Since the total number of votes is 571, you can set up the equation: \( x + (x + 89) = 571 \).

Step 4 - Simplify the Equation

Combine like terms: \( 2x + 89 = 571 \).

Step 5 - Solve for \( x \)

Subtract 89 from both sides to isolate the terms with \( x \): \( 2x = 482 \).

Step 6 - Divide to Find Value of \( x \)

Divide both sides by 2 to find \( x \): \( x = 241 \).

Step 7 - Calculate Winner's Votes

Add 89 to the loser's votes to find the winner's votes: \( 241 + 89 = 330 \).

Key Concepts

Variables Definition

Algebra word problems often require defining variables to represent unknown values. In this problem, we needed to find out how many votes each candidate received. To do this:

• Define the variable for the one we don't know directly: the loser's votes. Let's call this variable \( x \).
• Express the other quantity in terms of this variable: the winner's votes as \( x + 89 \).
  This helps to connect the two values based on the given relationships.

Linear Equations

Linear equations are fundamental in solving algebra word problems. They involve setting up an equation that represents the given conditions. Here’s how we set up the linear equation in this problem:

• Total votes equation: The problem states there were 571 total votes. The sum of votes for the winner and the loser should equal 571.
  So, we write: \( x + (x + 89) = 571 \).
• Simplify: Combine like terms to simplify the equation: \( 2x + 89 = 571 \).
• Solve: Isolate the variable by performing algebraic operations (subtract and divide) to find \( x \).

Problem-Solving Steps

Solving algebra word problems follows a series of logical steps. Let’s recap the problem-solving process used here:

• Understand the problem: Carefully read to identify known values and relationships.
• Define the variables: Clearly articulate unknowns and their relationships.
• Set up the equation: Translate the problem's words into a mathematical equation.
• Simplify the equation: Combine like terms and perform operations to isolate the variable.
• Solve for the variable: Use algebraic techniques to find the value of the unknown.
• Calculate the final values: Substitute the found values back to determine other related quantities.

This logical approach ensures structured and understandable solutions, making it easier to grasp complex problems.