Problem 27 - Prealgebra

Question

In the 2004 presidential election, Georgia had 15 electoral votes. That was 19 votes fewer than the number of electoral votes in Texas. Write and solve an equation to find the number of electoral votes in Texas.

Step-by-Step Solution

Answer

Texas had 34 electoral votes in 2004.

Step 1: Identify the Known Values

We are given that Georgia had 15 electoral votes in the 2004 presidential election. Additionally, it is mentioned that Georgia's 15 electoral votes were 19 fewer than the number of electoral votes in Texas. We need to use these details to form an equation.

Step 2: Define the Variable

Let the variable \( x \) represent the number of electoral votes in Texas.

Step 3: Formulate the Equation

Use the relationship provided: since Georgia had 19 fewer votes than Texas, the equation can be set up as follows:\[ x - 19 = 15 \]

Step 4: Solve the Equation

To determine the value of \( x \), add 19 to both sides of the equation:\[ x - 19 + 19 = 15 + 19 \]\[ x = 34 \]

Step 5: Interpret the Result

We have found that \( x = 34 \), which means Texas had 34 electoral votes in the 2004 presidential election.

Key Concepts

Equation Solving

Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes the equation true. In this exercise, we aimed to find out how many electoral votes Texas had based on the number known from Georgia. To solve this problem, we formulated an equation using the given information.

• The known value was that Georgia had 15 electoral votes.
• Georgia’s votes were 19 fewer than those in Texas.
• This leads us to create the equation: \[ x - 19 = 15 \]

Equation solving often involves several steps:

• Setting up an equation using known relationships.
• Manipulating the equation to isolate the variable on one side. Here, adding 19 to both sides helped to find the value of \( x \). \[ x = 34 \]

The solution shows the logical steps to derive \( x \), emphasizing the importance of balancing both sides of the equation.

Variables in Algebra

Variables are a core component of algebra, representing unknown or changeable quantities. In our problem, we used a variable to denote the number of electoral votes for Texas.

• The variable \( x \) was introduced to represent what we did not know: Texas's votes.

Using variables allows flexibility and helps in forming equations that needed to be solved. They symbolize unknown quantities in equations:

• Variables can take any value, but we solve equations to find their specific values under given conditions.
• They also allow for general solutions in problems with more than one possible answer.

In this case, once \( 19 \) was added to both sides of the equation \[ x - 19 = 15 \], we solved for the variable to find that \( x = 34 \). This illustrates how assigning a variable helps in solving real-world problems like determining electoral votes.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are the building blocks of equations.

• The expression \( x - 19 \) was used to show that Texas had 19 more votes than Georgia.

Understanding expressions is crucial to forming equations and solving problems.

• Expressions can be simplified or rearranged to make equation solving easier.
• They represent relationships or operations involving variables.

Through the exercise, we saw how expressing the original problem in algebraic terms allowed us to systematically solve it. By understanding the expression \( x - 19 \), we formed the foundation for solving the equation, emphasizing the role of algebraic expressions in problem-solving.