Problem 16 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$7(x+8)-4=10$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -6\).

Step 1: Distribute the 7

To eliminate the parentheses, apply the distributive property to the term \[7(x+8)\].This involves multiplying 7 by each term inside the parentheses:\[7 \times x + 7 \times 8 = 7x + 56.\]

Step 2: Simplify the Equation

Substitute the distributed expression back into the equation:\[7x + 56 - 4 = 10.\]Now, simplify the left side by combining like terms. Subtract 4 from 56:\[7x + 52 = 10.\]

Step 3: Isolate the Variable on One Side

To get the term with \(x\) by itself, subtract 52 from both sides of the equation:\[7x + 52 - 52 = 10 - 52\]which simplifies to:\[7x = -42.\]

Step 4: Solve for x

To solve for \(x\), divide both sides of the equation by 7:\[\frac{7x}{7} = \frac{-42}{7}.\]This simplifies to:\[x = -6.\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions and equations. When we see a term being multiplied by a sum or difference within parentheses, the distributive property helps us "distribute" the multiplication across each term inside. This property is particularly useful for removing parentheses, making equations easier to manage.

**Understanding the Mechanism**
Let's look at the expression from our exercise:

• We have \(7(x + 8)\).

Using the distributive property, we multiply 7 by each term inside the parentheses separately:

• \(7 \times x = 7x\)
• \(7 \times 8 = 56\)

Combining these results, we rewrite the expression as \(7x + 56\). This is how the distributive property simplifies expressions, setting the stage for further solving.

In practice, this step makes calculations smoother and reduces the complexity of the problem at hand, enabling us to focus on isolating the variable.

Isolation of Variables

In algebra, the goal often is to find the value of the unknown variable, usually represented by \(x\). This involves isolating the variable on one side of the equation through a series of operations, leaving a numerical value alone on the other side.

**Steps to Isolate**
From our exercise after applying the distributive property, we simplify further to achieve this:

• We begin with the equation \(7x + 52 = 10\).
• To isolate \(7x\), subtract 52 from both sides: \(7x + 52 - 52 = 10 - 52\).
• This simplifies to \(7x = -42\).

Isolation clears other numbers or expressions interfering with the variable and helps us get closer to the variable's value.

Simplifying Expressions

Simplifying expressions is crucial for making equations easier to solve. It often involves combining like terms, reducing fractions, or reorganizing the terms to facilitate further simplification.

**Applying Simplification**
After distributing and isolating, we continue simplifying the expression:

• We simplify \(7x + 56 - 4\) by combining the constants 56 and -4 to get \(7x + 52\).
• Each step of simplification reduces the complexity, making calculating the value of \(x\) simpler.

Finally, with our example, we transform \(7x = -42\) into the isolated form of \(x\) by dividing both sides by 7:

• \(x = \frac{-42}{7}\), simplifying to \(x = -6\).

This process demonstrates how simplification, when done correctly, streamlines our path to the final solution, enabling us to solve equations effectively.