Problem 53 - Elementary Algebra

Question

Solve. $$ 5 x-2=3 x+6 $$

Step-by-Step Solution

Answer

x = 4

Step 1: Move Variables to One Side

To isolate the variable terms, subtract \(3x\) from both sides of the equation: \[ 5x - 3x - 2 = 6 \] This simplifies to: \[ 2x - 2 = 6 \]

Step 2: Move Constants to the Other Side

Add 2 to both sides of the equation to move the constant term: \[ 2x - 2 + 2 = 6 + 2 \] Simplifying this gives: \[ 2x = 8 \]

Step 3: Solve for x

Divide each side of the equation by 2 to solve for \(x\): \[ \frac{2x}{2} = \frac{8}{2} \] Thus: \[ x = 4 \]

Key Concepts

Solving Equations

When it comes to solving linear equations, the goal is to find the value of the variable that makes the equation true. Linear equations, like the one in this example, are equations of the first degree, meaning the variable is raised to the power of one. To solve the equation \(5x - 2 = 3x + 6\), you need to perform a series of algebraic operations that simplify and eventually isolate the variable.A linear equation can typically be solved in several standard steps:

• Move terms involving the variable to one side of the equation.
• Move constant terms to the opposite side.
• Isolate the variable by performing mathematical operations.

This logical approach ensures that you unravel the equation step by step, maintaining balance on both sides.

Algebraic Manipulation

Algebraic manipulation is a key skill in solving equations. It involves using operations like addition, subtraction, multiplication, and division to rearrange and simplify equations. Let's examine the first step in this process.Initially, you need to move the variable terms to one side. In our example, we subtract \(3x\) from both sides, yielding \(5x - 3x - 2 = 6\) which simplifies to \(2x - 2 = 6\). This step uses subtraction to eliminate the variable on one side, allowing you to focus on solving for \(x\) on the other.Next, algebraic manipulation involves handling the constant terms. By adding 2 to both sides, we remove the constant from the left, simplifying the equation further to \(2x = 8\). This step makes it easier to isolate the variable, preparing you to solve the equation in the next step.

Isolating the Variable

Isolating the variable is a critical goal in solving linear equations, as it gives you the solution directly. In our equation \(2x = 8\), isolating \(x\) involves a simple division.To isolate \(x\), divide both sides of the equation by 2, which gives \(\frac{2x}{2} = \frac{8}{2}\). This results in \(x=4\), providing the value of the variable that satisfies the original equation.The art of isolating the variable involves ensuring that the variable stands alone on one side of the equation. This often requires reverse operations; if a variable is multiplied by a number, you divide. If a number is added, you subtract. These inverse operations are what finally reveal the value of the unknown variable in an equation.