Problem 60 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ x-4=-6 $$

Step-by-Step Solution

Answer

\(x = -2\)

Step 1: Identify the Equation

The given equation is \(x - 4 = -6\). We need to solve for \(x\) by isolating it on one side of the equation.

Step 2: Add 4 to Both Sides

To isolate \(x\), add \(4\) to both sides of the equation: \[ x - 4 + 4 = -6 + 4 \] Simplifying this gives: \[ x = -2 \]

Step 3: Verify the Solution

Substitute \(x = -2\) back into the original equation to check if it holds true: \[ (-2) - 4 = -6 \] Simplify the left side: \[ -6 = -6 \] Since both sides are equal, \(x = -2\) is the correct solution.

Key Concepts

Isolating Variables

In the context of solving linear equations, isolating variables is a fundamental process. This involves rearranging the equation to get the variable of interest by itself on one side of the equation. This is crucial because it simplifies the problem and allows us to find the solution for the variable directly.

To isolate a variable, we reverse any operations affecting it. For example, in the equation \(x - 4 = -6\), the variable \(x\) is being subtracted by 4. To isolate \(x\), we add 4 to both sides of the equation. This neutralizes the -4 on the left:

• The equation \(x - 4 = -6\) becomes \(x - 4 + 4 = -6 + 4\).
• As a result, \(x = -2\) because the -4 and +4 cancel each other out.

Isolating variables by undoing operations is a key step that allows deeper understanding and mastery of solving equations.

Checking Solutions

Once a solution to an equation is found, it is necessary to verify that the solution satisfies the original equation. This step ensures our answer is correct and reliable. To check a solution, substitute the found value back into the original equation.

For the equation \(x - 4 = -6\), we found that \(x = -2\). When we substitute -2 back into the equation, we perform:

• Replace \(x\) with -2: \((-2) - 4\).
• Simplify the expression: \(-6\).

We see that the left side equals the right side of the original equation, which is \(-6\). This confirms that \(x = -2\) is the correct solution. Always remember to check your solutions, as it confirms the accuracy of your work.

Equation Simplification

Equation simplification is about reducing an equation to its simplest form, making it easier to identify solutions. This process not only involves isolating variables but also performing basic arithmetic operations to tidy up the equation.

Consider the equation \(x - 4 = -6\). Once we've isolated \(x\) by adding 4 to both sides, we simplify further by performing the operation \(-6 + 4\), which results in \(-2\).

• Simplifying \(x - 4 + 4 = -6 + 4\) leads us to \(x = -2\).
• This simplification helps us to clearly see the solution without additional complexity.

Simplification ensures that errors are minimized and solutions are verified effectively. It is a crucial step that supports better understanding and easier calculations.