Problem 28 - College Algebra

Question

Solve for the variable. $$ 8(x+3)=64 $$

Step-by-Step Solution

Answer

x = 5

Step 1: Distribute the 8

First, we will apply the distributive property to the equation. This means multiplying the 8 by each term inside the parentheses. So, multiply 8 by both x and 3: \[ 8 \times x + 8 \times 3 = 8(x + 3) \]This simplifies to:\[ 8x + 24 = 64 \]

Step 2: Isolate the 8x

Next, we need to get all terms involving x on one side of the equation. We will do this by subtracting 24 from both sides of the equation to isolate the term with x:\[ 8x + 24 - 24 = 64 - 24 \]This simplifies to:\[ 8x = 40 \]

Step 3: Solve for x

Now that we have 8x isolated, we want to solve for x. To do that, divide both sides of the equation by 8:\[ \frac{8x}{8} = \frac{40}{8} \]This simplifies to:\[ x = 5 \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions and equations. It allows you to multiply a single term across terms inside parentheses.

In the equation given, \[8(x+3)=64\], we employ the distributive property in the first step to eliminate the parentheses. By multiplying 8 by each term inside \((x + 3)\), the expression becomes:

• \(8 \times x = 8x\)
• \(8 \times 3 = 24\)

These calculations transform the equation into: \[8x + 24 = 64\]This simplification makes it easier to continue solving the equation. Understanding and applying the distributive property efficiently can significantly enhance your ability to manipulate and solve algebraic expressions.

Equation Solving

Equation solving in algebra involves finding the value of the variable that makes the equation true. It requires simplifying the equation step by step to isolate the variable.

From the equation \(8x + 24 = 64\), the goal is to transform it into a simpler form where the variable is on one side.

By identifying like terms and performing basic arithmetic operations, we move towards a solution.

• First, subtract 24 from both sides of the equation to maintain equality: \[8x + 24 - 24 = 64 - 24\]
• This simplifies to: \[8x = 40\]

Each operation brings us closer to the simplest form possible. The key to successful equation solving lies in understanding the balance and keeping the equation equal on both sides throughout this process.

Variable Isolation

Variable isolation is the final goal in solving algebraic equations. It involves leaving the variable on one side and having a number or simplified expression on the other.

After distributing and simplifying, the equation \(8x = 40\) is ready for variable isolation.

To isolate \(x\), we divide both sides by 8, which is the coefficient of \(x\):

• \[\frac{8x}{8} = \frac{40}{8}\]

This operation simplifies the equation to \(x = 5\). At this stage, the value of the variable \(x\) is found.

Remember, the aim is to express \(x\) purely in terms of numbers, and doing so confirms that \(x = 5\) satisfies the original equation. Properly isolating variables is crucial as it reflects the solution to the problem in its simplest form.