Problem 13 - Introductory and Intermediate Algebra for College Students 4th

Question

Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. A number increased by 5 is two times the number. Find the number.

Step-by-Step Solution

Answer

The number is 5.

Step 1: Writing the equation

According to the problem, we know that a number (which we'll denote as \(x\)) increased by 5 is two times the number. This gives us the equation: \(x + 5 = 2x\)

Step 2: Simplifying the equation

We can simplify this equation by subtracting x from both sides to isolate the variable. This results in: \(5 = x\)

Key Concepts

Equation Solving

Solving equations is like solving a mystery; you need to find the value of an unknown number, often represented by a variable like \(x\). When solving linear equations, you're trying to find out exactly what this variable must be to make the equation true. In the equation "\(x + 5 = 2x\)", you're dealing with a simple linear equation.

To solve it, you need to follow a few basic steps:

• Write down the equation you need to solve.
• Use algebraic manipulations to isolate the unknown on one side of the equation.
• Simplify the equation until you find the solution.

In our example problem, this means figuring out what number, when increased by 5, equals twice the number itself.

This practice helps build a strong foundation for understanding more complex mathematical concepts later on.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. In the given exercise, "\(x + 5\)" and "\(2x\)" are examples of algebraic expressions. Both contain the variable \(x\), which stands for the unknown number we're trying to find.

Breaking these expressions down is crucial:

• \(x + 5\): This means the unknown number \(x\) is increased by 5.
• \(2x\): This states that the number \(x\) is being doubled.

Understanding how to analyze these expressions is key to forming and solving equations.
Once you write an equation from algebraic expressions, you follow systematic steps to simplify it, and eventually, find the value of the variable involved.

Variable Isolation

Variable isolation is an important skill when solving equations. It involves getting the variable you're trying to solve for all by itself on one side of the equation. In essence, you are 'isolating' the variable from other numbers or variables.

Let's apply it to the given problem: the equation \(x + 5 = 2x\).
To isolate \(x\), you'll want to:

• Subtract \(x\) from both sides of the equation to move all \(x\) terms to one side: \(x + 5 - x = 2x - x\).
• This simplifies to: \(5 = x\).

This tells us that \(x\) is equal to 5. Successfully isolating the variable \(x\) is what helps us find the solution.

Understanding how to isolate variables is essential because it sets the stage for working with more advanced algebraic techniques.