Problem 22 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$2 x+4=6$$

Step-by-Step Solution

Answer

The solution is \(x = 1\).

Step 1: Identify the equation

The given equation is \(2x + 4 = 6\). We need to isolate \(x\) on one side of the equation.

Step 2: Use the subtraction property of equality

To isolate \(x\), we first need to remove the constant from the left side. Subtract 4 from both sides of the equation: \(2x + 4 - 4 = 6 - 4\). This simplifies to \(2x = 2\).

Step 3: Solve for x using division

Divide both sides of the equation by 2 to solve for \(x\): \(\frac{2x}{2} = \frac{2}{2}\). This gives us \(x = 1\).

Key Concepts

Subtraction Property of Equality

The subtraction property of equality is a fundamental concept in algebra. When you have an equation, you can keep the two sides equal by performing the same subtraction on each side. This step is crucial when simplifying equations to isolate a variable.

For example, consider the equation \(2x + 4 = 6\). To isolate the term with the variable \(x\), we need to eliminate the constant on the left side. By using the subtraction property of equality, we subtract 4 from both sides:

• \(2x + 4 - 4 = 6 - 4\)

This simplifies the equation to \(2x = 2\).
This step is like balancing a scale — whatever you do to one side, you must do to the other to maintain balance. It helps maintain the integrity of the equation as you work towards isolating the variable.

Solving Linear Equations

Solving linear equations is all about finding the value of a variable that makes the equation true. In linear equations, variables are not squared, cubed, or otherwise raised to any power other than one. They form a straight line when graphed.

The key steps in solving a linear equation include:

• Eliminating constants on the same side as the variable with the use of the addition or subtraction property of equality.
• Simplifying the equation to get the variable by itself using division or multiplication if necessary.

For example, after using subtraction to simplify \(2x + 4 = 6\) to \(2x = 2\), the next step is to solve for \(x\) using division.
Linear equations frequently appear in mathematics, and mastering them is essential for solving more complex algebraic expressions. Understanding the process of isolating variables using basic operations allows you to solve these confidently.

Isolation of Variables

Isolation of variables is a vital process in solving equations. It involves rearranging the equation so that the variable you are solving for is on one side of the equation and everything else is on the other.

This process can involve several operations such as:

• Addition or subtraction to get rid of constants on the same side as the variable.
• Division or multiplication to simplify terms that have coefficients (numbers multiplied with the variable).

In the example of \(2x = 2\), we isolate \(x\) by dividing both sides by 2:
\(\frac{2x}{2} = \frac{2}{2}\), which simplifies to \(x = 1\).
This demonstrates isolating \(x\) by ensuring it stands alone on one side of the equation.
Isolation is often the ultimate goal in solving equations, as it provides the solution for the variable or variables you are interested in. The concept emphasizes reorganizing equations strategically by undoing operations applied to the variable step-by-step.