Problem 87 - Intermediate Algebra

Question

Solve each equation. $$ 2(2 x+1)=x+15+2 x $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 13 \).

Step 1: Expand the Equation

First, expand the left-hand side of the equation. Multiply the 2 with each term inside the parenthesis: \[ 2(2x + 1) = 4x + 2 \] Now the equation looks like this: \[ 4x + 2 = x + 15 + 2x \]

Step 2: Combine Like Terms

Now simplify the right-hand side by combining like terms: The terms \(x\) and \(2x\) combine to give \(3x\), so the equation becomes: \[ 4x + 2 = 3x + 15 \]

Step 3: Isolate the Variable on One Side

To solve for \(x\), we need to get all terms involving \(x\) on one side of the equation and constant terms on the other side. Subtract \(3x\) from both sides: \[ 4x - 3x + 2 = 15 \] This simplifies to: \[ x + 2 = 15 \]

Step 4: Solve for the Variable

Now, isolate \(x\) by subtracting 2 from both sides of the equation: \[ x + 2 - 2 = 15 - 2 \] This results in: \[ x = 13 \]

Key Concepts

Combining Like Terms

Understanding how to combine like terms is an essential skill in solving linear equations. This process involves simplifying expressions by adding or subtracting terms that have the same variable components. These terms are called "like terms." For example:

• In the equation \( 4x + 2 = x + 15 + 2x \), both \( x \) and \( 2x \) are like terms because they share the same variable \( x \).
• You can combine them by adding their coefficients: \( 1x + 2x = 3x \), simplifying the equation to \( 4x + 2 = 3x + 15 \).

Combining like terms helps you condense the equation, making it easier to manipulate and solve.
It reduces the complexity of the equation and gets you closer to finding the solution.

Isolating the Variable

Isolating the variable is a critical step in solving any equation and is all about getting our "x" alone on one side of the equation. In our example, once we've simplified the right side to \( 4x + 2 = 3x + 15 \), the next step is to bring all terms with \( x \) to one side and constants to the other. Here’s how you do it:

• You can start by subtracting \( 3x \) from both sides: \( 4x - 3x + 2 = 15 \). This leaves you with \( x + 2 = 15 \).
• Next, subtract 2 from both sides to further isolate the \( x \): \( x = 15 - 2 \).

These operations are meant to rearrange the equation into a form where \( x \) stands alone, allowing you to find its value.
Isolating the variable is especially important as it helps in clearly identifying the solution.

Distributive Property

The distributive property is a powerful tool in algebra that allows you to multiply a sum by distributing the factor across terms inside parentheses. It is expressed as: \[ a(b + c) = ab + ac \]In our original exercise, the use of the distributive property transforms the equation from \( 2(2x + 1) = x + 15 + 2x \) into \( 4x + 2 \) by multiplying 2 with both \( 2x \) and 1. Here’s what happens step-by-step:

• Multiply 2 with \( 2x \) to get \( 4x \).
• Multiply 2 with 1 to get 2.

This results in \( 4x + 2 \), which sets the basis for further simplifying the equation by combining like terms.
Understanding the distributive property enables you to handle equations with parentheses more effectively.