Problem 11 - Prealgebra

Question

Solve each equation. $$x+2=8$$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Understand the Equation

The given equation is \(x + 2 = 8\). Our goal is to solve for \(x\), which means we need to find the value of \(x\) that makes the equation true.

Step 2: Isolate the Variable

To isolate \(x\), we need to get rid of the \(+2\) next to \(x\). We can do this by subtracting 2 from both sides of the equation. This keeps the equation balanced. So, \(x + 2 - 2 = 8 - 2\).

Step 3: Simplify Both Sides

Simplifying both sides of the equation produces \(x + 0 = 6\), which is simply \(x = 6\). Now the value of \(x\) is isolated on one side of the equation.

Step 4: Verify the Solution

To ensure our solution is correct, substitute \(x = 6\) back into the original equation: \(6 + 2 = 8\). Since both sides of the equation are equal, our solution \(x = 6\) is correct.

Key Concepts

Prealgebra

Prealgebra is where many students start their journey in understanding algebraic concepts. It's a foundation that introduces you to numbers and operations that we build upon later. In Prealgebra, you learn to solve basic equations, which involves finding out what number a variable stands for. Let’s break this down simply.

• Think of the equation as a balance scale. Whatever you do to one side, you must do to the other to keep it balanced.
• Solving an equation like the one given is like a puzzle where you're finding the missing piece that makes the sides equal.
• The goal is to understand that variables like "x" represent unknown values that you aim to find.

It focuses on making these concepts intuitive so that more complex algebra won’t be intimidating later on.

Isolate Variable

Isolating the variable is one of the most crucial steps in solving an equation. It means you want to get the variable by itself on one side of the equation, which helps you see exactly what the variable equals. Here's how you can think about it:

• You have the equation: \(x+2=8\).
• To isolate \(x\), you need the '2' to disappear from the left side.
• Imagine you have to undo the addition of 2 to \(x\), so you subtract 2 from both sides.
• Keep balance—subtract the same thing from both sides: \(x+2-2=8-2\).
• This results in: \(x=6\).

This step is straightforward but incredibly important because isolating the variable allows for the easiest calculation, making it clear what the variable's value is.

Verify Solution

Once you've isolated the variable and found a solution, the next step is to verify that solution. This ensures that you didn't make a mistake and that your answer is indeed correct.

• Take the solution, which is \(x=6\), and substitute it back into the original equation.
• The original equation was \(x+2=8\). Substitute \(6\) for \(x\), making it \(6+2=8\).
• Check if the left side equals the right side: \(8=8\). It does!

Verification is a simple "plug and check" method. It's an assurance step that confirms the work you've done is correct and makes testing your results a habit. It helps build confidence in solving more complex equations as you advance in algebra.