Problem 46 - Prealgebra

Question

Simplify each side of the following equations first, then solve. $$8 a-6 a+a=8-14$$

Step-by-Step Solution

Answer

The solution is \(a = -2\).

Step 1: Simplify the Left Side

Start by combining the like terms on the left side of the equation: \(8a - 6a + a\). This simplifies to \(3a\) because \(8a - 6a + a = (8 - 6 + 1)a = 3a\).

Step 2: Simplify the Right Side

The right side of the equation is \(8 - 14\). Simplify this expression by subtracting: \(8 - 14 = -6\). So the equation after simplification is \(3a = -6\).

Step 3: Solve for a

Now that the equation is simplified to \(3a = -6\), divide both sides by 3 to isolate \(a\). This gives: \[a = \frac{-6}{3} = -2\].

Key Concepts

Simplifying Equations

Simplifying equations is the process of making an equation as clean and straightforward as possible. This involves performing operations that remove complexity without changing the equation's solution set. In practice, this often means reducing terms and getting rid of parentheses or combining constants. Here’s what you typically do to simplify an equation:

• Combine like terms (terms with the same variables raised to the same power).
• Perform operations like addition or subtraction if there are constant numbers on the same side.
• Clear any unnecessary parentheses by applying the distributive property, if needed.

For example, in our original equation, both sides have been simplified separately. The left side is simplified from several terms to just one, and the right side is simplified to a single number by subtraction.

Combining Like Terms

Combining like terms is a crucial skill in algebra, especially in prealgebra equations. Like terms are terms whose variables and their exponents (if any) are the same. Adding or subtracting these terms makes expressions simpler. Consider the expression:

• In the equation, we start with the left side: \(8a - 6a + a\).
• All of these terms are "like" because they all contain the variable \(a\).
• You combine them by adding or subtracting their coefficients: \(8 - 6 + 1 = 3\).

So, \(8a - 6a + a\) becomes \(3a\), simplifying the equation significantly.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. Once an equation is simplified, the next step is to isolate the variable to determine its value.For a simple linear equation like \(3a = -6\):

• Divide both sides of the equation by the coefficient of the variable. In this case, divide by 3 to solve for \(a\).
• Perform the division: \[a = \frac{-6}{3} = -2\]

This process yields the solution for the variable, and thus the equation is solved. By isolating the variable through division, you find that \(a\) equals \(-2\), and the solution is complete.