Problem 5 - Intermediate Algebra

Question

Solve each equation. \(-x-6=8\)

Step-by-Step Solution

Answer

The solution is \(x = -14\).

Step 1: Understanding the Given Equation

You are given the equation \[-x - 6 = 8\] The goal is to find the value of \(x\) that makes this equation true.

Step 2: Isolate the Variable Term

First, you want to move the constant term (-6) to the other side. To do this, add 6 to both sides of the equation:\[-x - 6 + 6 = 8 + 6\]Simplifying gives us:\[-x = 14\]

Step 3: Solve for x by Eliminating the Negative Sign

Now, we need to solve for \(x\). Since we have \(-x\), we can multiply both sides by -1 to eliminate the negative sign. This gives:\[(-1)(-x) = (-1)(14)\]Simplifying results in:\[x = -14\]

Step 4: Verify the Solution

Substitute \(x = -14\) back into the original equation to ensure it holds true:\[-(-14) - 6 = 8\]This simplifies to:\[14 - 6 = 8\]Since this is a true statement, our solution is correct.

Key Concepts

Isolating Variables

When solving linear equations, isolating the variable is a crucial first step. This process involves manipulating the equation to have the variable of interest on one side and constants on the other.

In our example equation \(-x - 6 = 8\), the goal is to isolate \(x\) on the left side. First, we start by eliminating any constants from the side of the variable. Here, adding 6 to both sides of the equation achieves that:

• Original equation: \(-x - 6 = 8\)
• Add 6 to both sides: \(-x - 6 + 6 = 8 + 6\)
• Simplify: \(-x = 14\)

This manipulation clears the constant term from the left side, allowing the focus to be on solving for the variable.

Negative Coefficients

Negative coefficients appear frequently in equations and can complicate the solution process. To make solving easier, it's essential to convert the negative coefficient to a positive one.

In the equation \(-x = 14\), there is a negative coefficient in front of \(x\). The simplest way to handle it is to multiply both sides of the equation by -1:

• \((-1)(-x) = (-1)(14)\)
• The result is: \(x = -14\)

Multiplying by -1 switches the sign, effectively turning the negative coefficient into a positive one. This small step makes it easier to work with and leads directly to finding the solution.

Equation Verification

Once a solution is found, verifying it is a good practice. This check ensures that no errors occurred during the solving process.

Verification involves plugging the found solution back into the original equation and checking if it results in a true statement. For this equation, substitute \(x = -14\) into the original equation \(-x - 6 = 8\):

• Substitute: \(-(-14) - 6\)
• Simplify: \(14 - 6 = 8\)

The result matches the original equation's outcome, confirming the solution is correct. Equation verification always solidifies confidence in the accuracy of your solution.