Problem 21 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$3 x-1=5$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2\).

Step 1: Understand the Equation

The given equation is \(3x - 1 = 5\). Our goal is to solve for \(x\).

Step 1: Add 1 to Both Sides

Use the addition property of equality to get rid of the \(-1\) on the left side. Add 1 to both sides of the equation:\[3x - 1 + 1 = 5 + 1\]This simplifies to:\[3x = 6\]

Step 2: Divide Both Sides by 3

Now, isolate \(x\) by dividing both sides by 3:\[\frac{3x}{3} = \frac{6}{3}\]This simplifies to:\[x = 2\]

Key Concepts

Addition Property of Equality

In mathematics, the addition property of equality is a fundamental concept used for solving equations. It states that if you have an equation, you can add the same number to both sides without changing the equation's balance. Let's break it down:

• If you have an equation such as \(a = b\), adding the same number \(c\) to both sides yields \(a + c = b + c\).
• This principle keeps the two sides equal and is vital for simplifying equations.

Think about a balance scale: if you add the same weight to both sides, the scale remains balanced. This analogy helps in understanding that the integrity of the equation is maintained. In our example \(3x - 1 = 5\), we use the addition property to eliminate \(-1\) by adding \(1\) to both sides, simplifying the equation to \(3x = 6\). With this step, we see firsthand how this property allows for an easier path to isolating the variable.

Algebra

Algebra is like a puzzle in mathematics where letters or symbols represent numbers. These letters are called variables, and they allow us to write equations and expressions that describe mathematical relationships. Here's a simple breakdown of key concepts:

• Variables: Usually letters, like \(x\) or \(y\), standing in place of unknown numbers.
• Constants: Numbers that have a fixed value in an equation.
• Operations: Symbols like \(+, -, \times, \div\) used to perform calculations on variables and constants.

In the equation \(3x - 1 = 5\), \(x\) is the variable. Algebra allows us to perform operations to isolate \(x\) and find its value. The beauty of algebra is in its ability to generalize and solve any equation that fits a similar pattern. This versatility makes algebra a powerful language of math.

Prealgebra

Prealgebra is the stepping stone to understanding algebra. It focuses on the basic mathematical concepts that provide a foundation for more advanced algebraic thinking. In prealgebra, you'll encounter topics such as:

• Understanding numbers and their properties, like integers and fractions.
• Learning how to perform basic operations on numbers.
• Introducing variables and simple equations.

Let's look at our equation \(3x - 1 = 5\). Prealgebra skills help us understand the operations involved—adding \(1\) to both sides—to isolate \(x\). It's all about breaking down an equation into manageable parts and understanding the steps needed to solve for an unknown. As you master prealgebra, you'll find yourself better equipped to handle more complex algebraic concepts.

Basic Arithmetic Operations

Arithmetic is the backbone of all of math, including algebra. Basic arithmetic operations include addition, subtraction, multiplication, and division. They are the building blocks of all mathematical calculations.Here's how these operations come into play:

• **Addition** and **subtraction** change numeric values and solve for variables by balancing equations.
• **Multiplication** is used when the same number should be added a particular number of times. For instance, \(3x\) signifies adding the value \(x\) three times.
• **Division** undoes multiplication. It simplifies equations by splitting one value into parts. By dividing \(3x\) by \(3\), we find the value of \(x\).

In our example, we used subtraction to rewrite \(3x - 1 = 5\) as \(3x = 6\), and then applied division to solve for \(x = 2\). Being adept at these operations helps in navigating through equations with ease.