Problem 123 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 8-3 a=32-2 a $$

Step-by-Step Solution

Answer

The solution is \(a = -24\).

Step 1: Arrange the Equation

We start by arranging the equation to have all terms involving the variable \(a\) on one side and constants on the other side. The given equation is \(8 - 3a = 32 - 2a\). To do this, add \(3a\) to both sides of the equation to move all \(a\) terms to one side. This gives us: \(8 = 32 + a\).

Step 2: Isolate the Variable

Now, we aim to isolate \(a\) on one side of the equation. Subtract 32 from both sides to move the constant to the other side. This results in: \(8 - 32 = a\), which simplifies to \(a = -24\).

Key Concepts

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. When we solve an equation, we're looking for that specific value. Think of it as solving a puzzle, where finding the missing piece is equivalent to identifying the variable's value. The equation given is a linear equation, meaning the variable is not raised to any power other than one.

Steps to solve linear equations generally include:

• Looking at both sides of the equation to understand how terms can be rearranged.
• Performing operations that help both sides of the equation to maintain equality.

Keeping these steps in mind helps in a systematic approach to finding the unknown variable. In our specific problem, we solve the equation by rearranging and isolating terms.

Variable Isolation

Variable isolation is the step where we focus on getting the unknown, in this case, the variable \(a\), all by itself on one side of the equation. This makes it clear what the value of the variable is.

We begin with the equation \(8 - 3a = 32 - 2a\). To isolate \(a\), notice how the terms with \(a\) are on opposite sides. By adding \(3a\) to both sides, we effectively eliminate the negative \(a\) term on the left side, simplifying to \(8 = 32 + a\).

Once \(a\) is isolated on one side, the equation becomes easier to solve. The next step to find \(a\) is to move all constant terms to the opposite side, ensuring \(a\) stands alone.

Equation Rearrangement

Equation rearrangement is the process of rewriting an equation in a form that makes it simpler to solve. By shifting terms around appropriately, we arrive at an equation where the variable can be isolated more easily.

We start with the equation \(8 - 3a = 32 - 2a\). In rearranging this equation, our goal is to bring all terms involving \(a\) together and all constant terms together. After moving the \(3a\) term to the right side, we have \(8 = 32 + a\).

Now, here's the part where clearer rearrangement took place. By subtracting \(32\) from both sides, you are rearranging the constants and the variable terms into distinct groups: \(8 - 32 = a\). It finally simplifies to \(a = -24\), which is the solution to our problem.