Problem 82 - Algebra for College Students

Question

Answer the question with an algebraic expression. The sum of two numbers is 65 , and one of the numbers is \(x\). What is the other number?

Step-by-Step Solution

Answer

The other number is represented by the expression \(65 - x\).

Step 1: Understand the Problem Statement

We are given that the sum of two numbers is 65, and one of these numbers is represented by the variable \( x \). We are asked to find the other number.

Step 2: Express the Relationship between the Numbers

Since the sum of the two numbers is 65, we can express this relationship with the equation: \( x + y = 65 \), where \( y \) is the other number we need to find.

Step 3: Solve for the Unknown Variable

To find the value of \( y \), rearrange the equation \( x + y = 65 \) to isolate \( y \). This can be done by subtracting \( x \) from both sides to get: \( y = 65 - x \).

Key Concepts

Equation Solving

Solving an equation involves finding the values of the variables that make the equation true. An equation is a mathematical statement that asserts the equality of two expressions. In this problem, we have the equation \( x + y = 65 \). This tells us that the sum of two numbers equals 65.
To solve the equation, we must manipulate it in such a way to find the value of the unknown variable, \( y \). Here are steps to solve such equations:

• Identify what is known and what needs to be found in the equation.
• Perform operations that will isolate the variable you are solving for.
• Check your solution by substituting it back into the original equation.

By following these steps, you ensure that your solution is consistent and accurate.

Algebra Fundamentals

Algebra teaches us how to use symbols and letters to represent numbers and quantities in equations and formulae. A fundamental part of algebra is understanding expressions and equations.
In an algebraic expression like \( x + y = 65 \), the symbol \( x \) is typically a variable that represents a number. The '=' sign signifies that the left side value is equal to the right side value. Breaking down this idea further:

• Variables are placeholders for numbers that can change or that are unknown.
• Equations state that two expressions are equal.
• Operations such as addition make up the structure of algebraic expressions.

Grasping these basics is important because they are the building blocks of algebraic problem-solving.

Variable Isolation

Variable isolation is crucial when you need to find the value of one variable in an equation. It involves rearranging the equation to have the variable on one side and everything else on the other.
In our example, to find \( y \), you subtract \( x \) from both sides of \( x + y = 65 \) which results in \( y = 65 - x \). Here's how you can approach variable isolation:

• Identify the variable you need to isolate.
• Use inverse operations to move other terms away from this variable.
• Perform the same operation on both sides of the equation to maintain balance.

This method lets you see clearly what one variable equals in terms of others, and helps solve the equation efficiently.