Problem 18 - Prealgebra

Question

Solve each equation. $$x-8=-3$$

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1: Identify the Objective

We need to find the value of \(x\) that makes the equation \(x - 8 = -3\) true.

Step 2: Add 8 to Both Sides

To isolate \(x\), we should eliminate the \(-8\) on the left side by adding 8 to both sides of the equation. This yields: \(x - 8 + 8 = -3 + 8\).

Step 3: Simplify both Sides

Simplify both sides of the equation. The left side simplifies to \(x\), and the right side simplifies to \(5\), so the equation becomes \(x = 5\).

Key Concepts

Steps for Solving Equations

When it comes to solving linear equations, following a systematic approach can simplify the task. Clear steps for solving equations not only build a stronger foundation in mathematics but also make the process straightforward and organized. Here is how you should approach it:

• Understand the Problem: Begin by examining what the equation is asking you to do. Identify the variable you need to solve for and the operations involved.
• Isolate the Variable: The primary goal is to have the variable on one side of the equation. This often entails performing operations such as addition, subtraction, multiplication, or division.
• Check Your Solution: After finding the value of the variable, always substitute it back into the original equation to ensure that it satisfies the equation.

By following these steps, you can effectively tackle any basic linear equation.

Isolating the Variable

In any linear equation, isolating the variable is a key step. It means getting the variable by itself on one side of the equation. This process involves reversing the operations around the variable. Let's use the equation given: \(x - 8 = -3\).To isolate \(x\), we need to get rid of the \(-8\). Since \(-8\) is subtracted from \(x\), we do the opposite operation—addition. Therefore, we add 8 to both sides:

• Initial equation: \(x - 8 = -3\)
• Add 8 to both sides: \(x - 8 + 8 = -3 + 8\)

This process brings us a step closer to pinpointing the value of the variable. Remember, whatever you do to one side of the equation, you must do to the other side to maintain balance.

Simplifying Equations

Once you've isolated the variable, the next logical step is simplifying the equation. This is where you tidy up and make sure everything is clear and concise. For the equation we are working with, simplifying means performing the arithmetic operations to eliminate any redundant terms:

• After adding 8 to both sides, simplify: \(x - 8 + 8\) simplifies to \(x\), and \(-3 + 8\) simplifies to \(5\).
• The equation thus becomes: \(x = 5\).

Simplification ensures that the equation is easy to interpret and that your solution is clear. It's important to double-check that every operation has been correctly carried out and that any unnecessary terms are removed. This attention to detail guarantees a clean and correct result.