Problem 28 - College Algebra

Question

For the following exercises, solve for the variable. $$ 8(x+3)=64 $$

Step-by-Step Solution

Answer

The solution is \( x = 5 \).

Step 1: Expand the Parentheses

First, expand the expression inside the parentheses by distributing the 8. This means you multiply both terms inside the parentheses by 8:\[ 8(x+3) = 8 \times x + 8 \times 3 \]This simplifies to:\[ 8x + 24 \]

Step 2: Set Up the Equation

After distributing, set up the equation with the expanded terms:\[ 8x + 24 = 64 \]

Step 3: Isolate the Term with the Variable

To isolate the term with the variable, subtract 24 from both sides of the equation:\[ 8x + 24 - 24 = 64 - 24 \]This simplifies to:\[ 8x = 40 \]

Step 4: Solve for the Variable

Divide both sides of the equation by 8 to solve for \( x \):\[ \frac{8x}{8} = \frac{40}{8} \]This simplifies to:\[ x = 5 \]

Key Concepts

Distributive Property

The distributive property is an essential algebraic concept that helps simplify expressions and solve equations. It's particularly useful when you have a term outside of parentheses that needs to be multiplied by each term within the parentheses. In this example, you see the distributive property applied to the equation \(8(x + 3) = 64\). To use the distributive property, multiply the number outside the parentheses, which is 8, by each term inside the parentheses:

• First, multiply 8 by \(x\) to get \(8x\).
• Next, multiply 8 by 3 to get 24.

After applying the distributive property, the expression \(8(x + 3)\) simplifies to \(8x + 24\). This step makes it easier to work with and solve the equation. Remember, the distributive property allows you to "distribute" a multiplication over addition or subtraction within parentheses.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. For the equation \(8x + 24 = 64\), our goal is to determine the value of \(x\). Once you've applied the distributive property and gotten \(8x + 24 = 64\), you're set to solve the equation.

• First, you'll need to ensure that all terms containing the variable are on one side of the equation. In this scenario, \(8x\) is already isolated on the left side.
• Next, to further simplify, you aim to "clean up" the equation by removing the constant term from the left side (here, the 24). You do this by subtracting 24 from both sides, which gives you \(8x = 40\).

By following these steps, you refine the problem into a simpler equation. Solving linear equations often involves carefully restructuring terms to uncover the value you're solving for.

Isolation of Variables

Isolation of variables is a crucial part of solving equations. It refers to getting the variable we're solving for by itself on one side of the equation. Once we reach the equation \(8x = 40\), our focus is isolating \(x\). We do this through division, as multiplication affects \(x\) in the equation:

• To isolate \(x\), divide both sides of the equation by 8. This action cancels out the 8 on the left side, leaving \(x\) alone.
• The resulting calculation is \(\frac{8x}{8} = \frac{40}{8}\), which simplifies directly to \(x = 5\).

By isolating the variable, you effectively solve the equation, finding that \(x = 5\) satisfies the original equation. This process is not only helpful in algebra but also forms the foundation for more complex mathematical topics.