Problem 39 - Prealgebra

Question

The following equations contain parentheses. Apply the distributive property to remove the parentheses, then simplify each side before using the addition property of equality. $$2(x+3)-x=4$$

Step-by-Step Solution

Answer

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

Step 1: Apply the Distributive Property

Start by applying the distributive property to remove the parentheses on the left side of the equation. Multiply each term inside the parentheses by 2: \[ 2(x+3) = 2x + 6 \] So, the equation becomes: \[ 2x + 6 - x = 4 \]

Step 2: Simplify Both Sides

Combine like terms on the left side of the equation. \[ 2x - x + 6 = x + 6 \] Thus, the equation simplifies to: \[ x + 6 = 4 \]

Step 3: Solve the Equation

Use the addition property of equality to isolate the variable \( x \). Subtract 6 from both sides of the equation: \[ x + 6 - 6 = 4 - 6 \] Which simplifies to: \[ x = -2 \]

Key Concepts

Understanding the Distributive Property

The distributive property is an essential concept in algebra that helps in simplifying expressions and solving equations. It allows you to remove parentheses in expressions where a number or variable multiplies a sum or difference within parentheses. Let's consider the exercise equation \(2(x+3)-x=4\).

The distributive property tells us that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. Thus:

• When you see \(2(x+3)\), you distribute the 2 to both \(x\) and the 3. This results in \(2x + 6\).
• This transformation is based on \(a(b+c) = ab + ac\), where each term inside the parentheses is independently multiplied by the outside factor.

Applying this property helps to eliminate parentheses, which is a crucial step in simplifying and solving equations.

Simplification Techniques

Simplification is the process of reducing an expression to its most concise form, making it easier to solve. After applying the distributive property in our equation, we end up with \(2x + 6 - x = 4\).

The next step is to simplify the left side of the equation by combining like terms:

• "Like terms" are terms that have identical variable parts. For example, in \(2x + 6 - x\), both \(2x\) and \(-x\) are like terms as they both involve the variable \(x\).
• Combine these terms to simplify: \(2x - x = x\).
• Now, the equation reduces to \(x + 6 = 4\).

This simplification makes the equation much more manageable and ready for the next solving steps.

Applying the Addition Property of Equality

Once the expression is simplified, the addition property of equality is used to isolate the variable and solve the equation. This property states that you can add or subtract the same number from both sides of the equation, and the equation remains balanced.

In the equation \(x + 6 = 4\), you want to solve for \(x\). Here's how you can apply the addition property of equality:

• Subtract 6 from both sides: \(x + 6 - 6 = 4 - 6\).
• This simplifies to \(x = -2\), isolating the variable \(x\).

This property helps in maintaining equality by ensuring that the solution remains correct as you solve for \(x\). Understanding and applying it correctly is crucial in solving linear equations.