Problem 31 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$2 x-4=-20$$

Step-by-Step Solution

Answer

The value of \(x\) is \(-8\).

Step 1: Understand the Problem

We are given the equation \(2x - 4 = -20\). The goal is to solve for \(x\) using the addition property of equality, which states that you can add the same number to both sides of an equation without changing the equality.

Step 2: Apply the Addition Property

The equation is \(2x - 4 = -20\). To isolate the term with \(x\), we first need to eliminate the \(-4\). Apply the addition property by adding \(4\) to both sides of the equation to maintain equality: \[2x - 4 + 4 = -20 + 4\] This simplifies to: \[2x = -16\].

Step 3: Solve for x

Now that the equation is simplified to \(2x = -16\), we need to solve for \(x\). Divide both sides by \(2\) to isolate \(x\): \[\frac{2x}{2} = \frac{-16}{2}\] This gives us: \[x = -8\].

Step 4: Verify the Solution

To ensure our solution is correct, substitute \(x = -8\) back into the original equation: \[2(-8) - 4 = -20\] Simplifying the left side: \[-16 - 4 = -20\] Since both sides are equal, the solution is verified.

Key Concepts

Understanding the Addition Property of Equality

When working with linear equations, the addition property of equality is a basic tool that is particularly useful. This property states that if you add (or subtract) the same value from both sides of an equation, the equation remains balanced. For example, consider the equation: - If you have something like \( a = b \), you can add, say, \( c \) to both sides and the equation will still hold true: \( a + c = b + c \).
This property is invaluable when it comes to simplifying equations and isolating variables. In the given exercise of \( 2x - 4 = -20 \), applying the addition property allowed us to remove the \( -4 \) from the left side by adding \( 4 \) to both sides, simplifying our equation to \( 2x = -16 \).
Being comfortable with the addition property is essential as it lays the foundation for solving more complex equations.

Isolation of Variables

A core goal in solving linear equations is to isolate the variable you're solving for, often denoted as \( x \). The process of isolation means systematically performing operations that "release" the variable from other terms until it stands alone on one side of the equation.
In our problem, after applying the addition property of equality, we reduced \( 2x - 4 = -20 \) to \( 2x = -16 \). Here, \( x \) is still multiplied by \( 2 \), so to isolate \( x \), we divide every term by \( 2 \): \[ \frac{2x}{2} = \frac{-16}{2} \]
This simplifies to \( x = -8 \). Successfully isolating \( x \) is a pivotal step in finding your solution.

Verifying Solutions

Once you've found a solution in an equation, it's crucial to ensure its accuracy. Verification is all about checking if your solution truly satisfies the original equation.
For our problem, with \( x = -8 \), we substitute this value back into the original equation: \[ 2(-8) - 4 = -20 \] Translating this to numbers, the left-hand side equates to \(-16 - 4 \), which simplifies to \(-20 \). Since both sides equal \( -20 \), our solution is confirmed as correct.
This step is not just a formality. It provides confidence that no error has crept into the calculations, thus reinforcing the reliability of your solution process. Always strive to make this final verification a habit.