Problem 40 - Elementary Algebra

Question

Solve. $$ 8 x-5=3 x 2 $$

Step-by-Step Solution

Answer

The solution for \(x\) is \(x = \frac{5}{2}\).

Step 1: Simplify Both Sides

First, simplify both sides of the equation. The right side, \(3x2\), should be simplified to \(6x\). This gives us a new equation: \(8x - 5 = 6x\).

Step 2: Get All \(x\) Terms on One Side

Subtract \(6x\) from both sides of the equation to get all \(x\) terms on one side. This yields \(8x - 6x - 5 = 0\) which simplifies to \(2x - 5 = 0\).

Step 3: Isolate the Variable Term

Add 5 to both sides to isolate the \(2x\) term. This results in the equation \(2x = 5\).

Step 4: Solve for \(x\)

Divide both sides of the equation by 2 to solve for \(x\). This results in \(x = \frac{5}{2}\).

Key Concepts

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. It is a fundamental skill in algebra. Imagine an equation as a balance scale. You want to keep both sides equal, while working to simplify the equation.

• First, you ensure every term is clearly defined. For this exercise, we simplify the right side by calculating \(3 \times 2\) to get \(6\).
• The goal is to manipulate the equation to isolate the variable we're solving for. In this context, it's \(x\).

To solve the equation \(8x - 5 = 6x\), we need to use methods that maintain balance. This includes simplifying both sides, relocating terms, and ultimately isolating variables.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations using arithmetic operations. This skill is key in making equations easier to work with.

• Begin by simplifying terms with multiplication, such as \(3 \times 2\), which equals \(6\).
• To manipulate the equation \(8x - 5 = 6x\), we move all terms involving \(x\) to one side. We do this by subtracting \(6x\) from both sides, resulting in \(2x - 5 = 0\).

Throughout this process, ensuring each operation is done properly is vital to maintaining the integrity of the equation.

Isolation of Variables

Isolation of variables means getting the variable by itself on one side of the equation. This step is crucial for determining its value.

• In our equation, \(2x - 5 = 0\), we need to free \(2x\) from surrounding numbers. Begin by adding \(5\) to both sides, producing \(2x = 5\).
• The final step requires dividing both sides by \(2\) to solve for \(x\), resulting in \(x = \frac{5}{2}\).

Each step carefully removes any interference from the variable, until it stands alone. This clarity is what allows us to accurately find its value.