Problem 38 - Prealgebra

Question

Simplify each side of the following equations before applying the addition property. $$10-6=8 x-7 x+6$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -2\).

Step 1: Simplify the Left Side

The left side of the equation is simple because it's a single constant expression: \( 10 - 6 \). Subtract to simplify it. \( 10 - 6 = 4 \). So, the left side simplifies to 4.

Step 2: Simplify the Right Side

The right side is \(8x - 7x + 6\). Combine the like terms, which are \(8x\) and \(-7x\). \(8x - 7x = 1x = x\). Thus, the right side becomes \(x + 6\).

Step 3: Set the Simplified Equation

Now, substitute the simplified expressions into the original equation: \(4 = x + 6\).

Step 4: Apply the Addition Property

To isolate \(x\), we need to get rid of the \(6\) on the right. Subtract \(6\) from both sides of the equation to balance it. \(4 - 6 = x + 6 - 6\) This simplifies to \(-2 = x\).

Key Concepts

Simplification of Expressions

Simplification is an essential skill in solving equations. It's like tidying up a room before you start a project. By simplifying, we reduce complex mathematical expressions into their most basic form, which makes them easier to work with.

In the given exercise, the left side of the equation starts with the expression \(10 - 6\). By subtracting, we simplify this to \(4\). On the right side, we have \(8x - 7x + 6\). Here, the simplification involves combining like terms (more on this later). The \'like terms\' are the expressions containing the variable \(x\).

• First, identify similar parts of the expression. This means finding terms that can combine to form simpler terms.
• Perform arithmetic operations, such as addition or subtraction, to condense these terms into a simpler form.

Simplifying expressions helps pave the way for applying other mathematical processes, such as solving equations efficiently.

Addition Property

The addition property of equality is a principle stating that you can add or subtract the same number from both sides of an equation, and it remains balanced. This property is crucial when you want to solve for a specific variable.

In our example, after simplifying, we end up with the equation \(4 = x + 6\). To solve for \(x\), we need to use the addition property. We aim to isolate \(x\) on one side of the equation. Here's how we apply it step-by-step:

• Subtract \(6\) from both sides of the equation: \(4 - 6 = x + 6 - 6\).
• This simplifies the equation to \(-2 = x\).

By using the addition property, we effectively remove the \(6\) from the right side, achieving the goal of solving for \(x\). This property is invaluable for maintaining the balance of an equation while manipulating its terms.

Combining Like Terms

Combining like terms is a useful method for simplifying expressions, particularly when dealing with equations involving variables. 'Like terms' are terms that have the same variables raised to the same power; these can be added or subtracted directly from each other.

Consider the expression \(8x - 7x + 6\) from the exercise. The terms \(8x\) and \(-7x\) are 'like terms' because both contain the variable \(x\).

• Combine these like terms by performing the respective arithmetic operation: \(8x - 7x = 1x = x\).
• Once combined, you simplify the right side of the equation to \(x + 6\).

By merging like terms, you simplify the equation, making it simpler to reach a solution. This step is crucial in reducing the complexity and number of terms you work with, particularly in equations involving multiple variables or complicated terms.