Problem 79 - Algebra A Combined Function

Question

Solve each equation. $$ x-5=0 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 5 \).

Step 1: Isolate the Variable

Our equation is \( x - 5 = 0 \). The goal is to solve for \( x \). First, add 5 to both sides of the equation to isolate \( x \). So we have \( x - 5 + 5 = 0 + 5 \).

Step 2: Simplification

This simplifies to \( x = 5 \). Now \( x \) is by itself, and we have found the value of \( x \).

Step 3: Verification

To double-check, substitute \( x = 5 \) back into the original equation: \( 5 - 5 = 0 \). Since the equation holds true, our solution \( x = 5 \) is correct.

Key Concepts

Isolation of Variables

The isolation of variables is the process of getting the variable by itself on one side of the equation. This is a critical first step in solving linear equations. By isolating the variable, we can determine its value by making everything else disappear from that side.

To isolate the variable, we perform algebraic operations that simplify the equation while keeping it balanced. Think of it as balancing a scale—what you do to one side, you must do to the other.

• Identify the term with the variable. In our example, it's the "x" in the equation x - 5 = 0.
• Decide what needs to be removed to isolate the variable. Here, the "-5" needs to go.
• Add 5 to both sides to cancel out the -5: x - 5 + 5 = 0 + 5.
• This manipulation leaves us with x = 5, nicely isolated.

By carefully performing these operations, the variable emerges clearly, ready for the next simplification step.

Simplification in Algebra

Simplification in algebra involves reducing expressions to their most basic form. Once we've isolated the variable, it's time to simplify the equation to reveal the variable's value. Simplifying makes equations more manageable and solutions more evident.

In the given equation, once the variable is isolated (as in x = 5), our work is nearly done. But simplification doesn't just occur at the end; it can also mean tidying up all the terms during the process of solving.

• Evaluate changes: After adding 5 to both sides, look at what remains: x = 5.
• Ensure there's no unnecessary clutter: The equation has become simpler, free of other operations or terms.
• The outcome is a clear, direct solution statement: x = 5.

Simplification transforms equations into easier forms to handle, allowing us to focus on what's essential—and that's knowing the variable's value.

Verification of Solutions

Verification of solutions is an essential final step in solving equations. This process confirms that our solution is correct and that no mistakes have been made. By verifying, we can be confident in the accuracy of our answer.

For verification:

• Substitute the found value back into the original equation. Here, replace x with 5 in x - 5 = 0.
• Perform the operation: 5 - 5 = 0.
• Check if both sides of the equation are equal. They are, indicating our solution is correct.

Verification provides peace of mind, ensuring that the steps you followed led to the right outcome. It's like double-checking your math homework, a quick confirmatory glance that can catch potential errors and assure your confidence in mathematical reasoning.