Problem 48 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ \frac{1}{3}(3 x-12)=6-2(x-1) $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 4 \).

Step 1: Simplify Both Sides of the Equation

Multiply \( \frac{1}{3} \) by each term inside the parenthesis on the left side of the equation as follows: \( \frac{1}{3} (3x - 12) = x - 4 \). Likewise, distribute the '2' on the right side of the equation as such: \( 6 - 2(x - 1) = 6 - 2x + 2 = 8 - 2x \).

Step 2: Organize the Equation

Move all terms involving 'x' to one side and the constant term to the other. This is done by adding \( 2x \) to both sides, and then adding '4' to both sides. The equation becomes: \( x + 2x = 8 + 4 \) that gives \( 3x = 12 \)

Step 3: Solve for 'x'

Finally, divide both sides by '3' to isolate 'x': \( \frac{3x}{3} = \frac{12}{3} \) or \( x = 4 \)

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial step in solving any linear equation. It involves breaking down complex expressions into easier, more manageable forms. When dealing with an equation such as \( \frac{1}{3}(3x - 12) = 6 - 2(x-1) \), our goal is to simplify both sides of the equation.

• Left Side: The expression \( \frac{1}{3}(3x-12) \) can be simplified by multiplying each term inside the parenthesis by \( \frac{1}{3} \). This results in simplifying to \( x - 4 \).
• Right Side: Similarly, simplify \( 6 - 2(x-1) \) by distributing the \( -2 \) across \( (x-1) \), which simplifies to \( 8 - 2x \) after also adding \( 6 + 2 \).

By simplifying expressions effectively, you ensure that the equation is in its most manageable form for the next steps of solving.

Distributive Property

The distributive property is a critical algebraic property that allows you to simplify equations by distributing a multiplier outside the parentheses to each term inside the parentheses. This property states \( a(b + c) = ab + ac \). In the context of the example equation \( \frac{1}{3}(3x - 12) = 6 - 2(x-1) \), we see the distributive property in action.

• On the Left: Applying \( \frac{1}{3} \) to both \( 3x \) and \( -12 \) within the parentheses gives \( x - 4 \).
• On the Right: The term \( -2(x - 1) \) is simplified by distributing \( -2 \) to both \( x \) and \( -1 \), which changes the expression to \( -2x + 2 \).

Understanding how the distributive property works simplifies complex expressions and is essential when rearranging terms within an equation.

Isolating Variables

Isolating variables is a fundamental technique to solve an equation and find the value of the unknown variable. The main goal is to get the variable, typically 'x', by itself on one side of the equation, resulting in a solution. Once you've simplified both sides of the equation, as was shown in the problem \( x - 4 = 8 - 2x \), the next step is to isolate 'x'.

• Start with gathering all x terms on one side: Add \( 2x \) to both sides to get \( x + 2x = 8 + 4 \), simplifying to \( 3x = 12 \).
• Next, isolate 'x' completely: Divide both sides by 3 to achieve \( x = 4 \).

This process of moving terms and systematically isolating the variable helps in achieving the solution efficiently and effectively.