Problem 17 - Prealgebra

Question

Solve each equation. $$x-5=-4$$

Step-by-Step Solution

Answer

The solution is \( x = 1 \).

Step 1: Understanding the Equation

We begin with the equation \( x - 5 = -4 \). Our goal is to find the value of \( x \) that satisfies this equation.

Step 2: Isolate the Variable

To isolate the variable \( x \), we need to eliminate the \( -5 \) from the left-hand side. We do this by adding \( 5 \) to both sides of the equation.

Step 3: Perform the Addition

Add \( 5 \) to both sides of the equation: \( x - 5 + 5 = -4 + 5 \). Simplifying both sides gives \( x = 1 \).

Step 4: Verify the Solution

Substitute \( x = 1 \) back into the original equation to verify: Does \( 1 - 5 = -4 \)? Yes, this is true, which confirms our solution.

Key Concepts

Isolate the Variable

In solving linear equations, one of the first and most crucial steps is to isolate the variable. This process involves rearranging the equation to get the unknown variable by itself on one side of the equation. Let's look at how this was accomplished in the exercise:- **Identify the operations affecting the variable:** In the equation \( x - 5 = -4 \), the variable \( x \) is subtracted by 5.- **Do the opposite operation:** To isolate \( x \), we need to undo the subtraction by doing the opposite operation — which is addition.
In this context, you add 5 to both sides of the equation to maintain balance, leading us to \( x - 5 + 5 = -4 + 5 \).- **Simplify the equation:** This simplifies to \( x = 1 \), where \( x \) stands alone on one side of the equation.When isolating the variable:- Always do the same operation to both sides of the equation.- Balance is crucial — whatever you do to one side, do it to the other.- Simplify the equation step-by-step, focusing on one operation at a time.Isolating the variable is essential in finding the most direct path to the solution.

Verify the Solution

After finding a potential solution, the next critical step is to verify it. Verifying your solution ensures that your answer truly satisfies the original equation:- **Substitute the found value back into the original equation:** In our example, we substitute \( x = 1 \) back into \( x - 5 = -4 \).
This results in checking: \( 1 - 5 = -4 \) which holds true.- **Consistency check:** Verifying confirms the accuracy of the solution and helps catch any mistakes from prior steps. **Why verify?**- Ensures the solution satisfies the original condition.- Helps in understanding the relationship between the numbers involved in the equation.Verification is a powerful step that provides confidence in your result and is a good practice in all mathematical problem-solving.

Prealgebra Concepts

Prealgebra serves as the foundation for understanding basic algebraic principles. It introduces fundamental operations that are crucial in solving equations:- **Understanding operations:** Addition, subtraction, multiplication, and division lay the groundwork for algebra.- **Arithmetic fluency:** Ability to perform operations quickly and accurately helps when isolating variables and verifying solutions.**Key Prealgebra Concepts:**- **Solving for unknowns:** Often the core of prealgebra exercises involves finding what we don't know (the variable) based on what we do know.
In the exercise, solving for \( x \) uses basic arithmetic to isolate the variable.- **Number sense:** Understanding positive and negative numbers is crucial.
Knowing \( -5 \, + \, 5 = 0 \) was key in simplifying the equation to \( x = 1 \).Prealgebra is all about building intuition with numbers and operations, fostering skills needed for more complex algebra.