Problem 82 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$3(x-2)=6$$

Step-by-Step Solution

Answer

The solution is \( x = 4 \).

Step 1: Distribute the Coefficient

Start by applying the distributive property to eliminate the parentheses in the equation. Multiply 3 by each term inside the parentheses: \[ 3(x - 2) = 3 \cdot x + 3 \cdot (-2) = 3x - 6 \] So the equation now becomes: \[ 3x - 6 = 6 \]

Step 2: Isolate the Variable Terms

Add 6 to both sides of the equation to move the constant term to the right side of the equation. This will help in isolating the variable: \[ 3x - 6 + 6 = 6 + 6 \] Simplify to get: \[ 3x = 12 \]

Step 3: Solve for x

Divide both sides by 3 to solve for the variable, \( x \): \[ \frac{3x}{3} = \frac{12}{3} \] Simplify to find: \[ x = 4 \]

Key Concepts

Distributive Property in Equations

The distributive property is a crucial algebraic rule that helps simplify expressions and solve equations. It involves removing parentheses by distributing a multiplication over addition or subtraction within the parentheses. In essence, when you have an expression like \( a(b + c) \), you apply the distributive property by multiplying \( a \) with both \( b \) and \( c \).
In the exercise \( 3(x-2)=6 \), the problem initially presents an equation where the distributive property is employed. Here's how that works:

• Multiply the 3 by the \( x \), giving you \( 3x \).
• Multiply the 3 by the \(-2\), resulting in \(-6\).

This results in the equation transforming from \( 3(x-2) \) to \( 3x - 6 \).
Using the distributive property greatly simplifies equations, making them easier to manage and solve. It's a fundamental step in moving forward with more complex algebraic expressions.

Variable Isolation

Variable isolation is a key step in solving equations, especially linear ones. The goal is to get the variable of interest by itself on one side of the equation. This makes it possible to easily see what number the variable equates to.
After using the distributive property in the original problem, you are left with the equation \( 3x - 6 = 6 \). To isolate \( x \), you must eliminate any numbers hindering its separation from the constants. Here's what you need to do:

• Add 6 to both sides of the equation to remove the \(-6\) from the left: \( 3x - 6 + 6 = 6 + 6 \).

This simplifies to \( 3x = 12 \).
All that's left is to divide each side by 3, allowing you to solve for \( x \). This process exemplifies the power of maintaining balance and equality within equations for them to remain true as you solve.

Understanding Linear Equations

Linear equations are among the simplest types of equations and form the foundation of more complex algebra. They are called "linear" because when graphed on a coordinate plane, they produce a straight line.
An equation is considered linear if it can be written in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. These equations have no exponents higher than 1 attached to the variable, which makes them different from quadratic or polynomial equations.
The initial exercise presents a linear equation in the form \( 3(x - 2) = 6 \). Upon distribution, you have \( 3x - 6 = 6 \), which confirms it's linear as it adheres to the format \( ax + b = c \).
Linear equations usually involve straightforward steps of applying arithmetic operations—addition, subtraction, multiplication, division—to isolate the variable. They are essential in algebra for understanding relationships between variables and constants, laying the groundwork for solving more complex algebraic expressions.