Problem 18 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$5(2 x-4)+8=38$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Distribute

First, distribute the 5 into the parenthesis. Multiply 5 by each term inside the parenthesis:\[5(2x - 4) = (5 \times 2x) + (5 \times -4) = 10x - 20\]

Step 2: Simplify Equation

Now, substitute the distributed expression back into the original equation:\[10x - 20 + 8 = 38\]Simplify the left side by combining like terms:\[10x - 12 = 38\]

Step 3: Isolate the Variable Term

Add 12 to both sides of the equation to isolate the term with the variable:\[10x - 12 + 12 = 38 + 12\]This simplifies to:\[10x = 50\]

Step 4: Solve for x

Divide both sides by 10 to solve for \(x\):\[\frac{10x}{10} = \frac{50}{10}\]This simplifies to:\[x = 5\]

Key Concepts

Understanding the Distributive Property

The distributive property is a powerful and essential tool in algebra that enables us to simplify expressions and solve equations. It involves distributing a multiplication operation over addition or subtraction within parentheses.

This means taking a term outside the parentheses and multiplying it by each term inside. In the given exercise, the expression \(5(2x - 4)\) requires us to distribute the 5 to both \(2x\) and -4.

• Multiply 5 by \(2x\) to get \(10x\).
• Then, multiply 5 by -4 to get -20.

These results are added together, resulting in the expression \(10x - 20\). The distributive property helps simplify expressions by eliminating parentheses, paving the way for further simplification steps.

Combining Like Terms for Simplification

After using the distributive property, the next step is to combine like terms to simplify the equation. Like terms are terms that contain the same variable raised to the same power. In our exercise, once the expression is distributed, we get \(10x - 20 + 8\).

Notice the constants -20 and +8. These are like terms because they have no variable attached, and they can be combined:

• Combine the constants: -20 + 8 to get -12.
• The expression then simplifies to \(10x - 12 = 38\).

This simplification is crucial as it helps to make the equation more straightforward, setting it up for solving by isolating the variable.

Isolation of Variables to Solve Equations

The final concept in solving linear equations is isolating the variable, which allows us to find its value. The expression needs to be rearranged so that the variable is by itself on one side of the equation.

After simplifying to \(10x - 12 = 38\), we need to focus on the term \(10x\). The isolation process involves:

• Adding 12 to both sides, canceling out the -12: \(10x - 12 + 12 = 38 + 12\), simplifying to \(10x = 50\).
• Dividing each side by 10 to solve for \(x\): \(x = 5\).

By performing these operations, we systematically isolate \(x\), allowing us to solve for it efficiently while maintaining balance in the equation.