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Chapter 2: Problem 3

Solve the equations by using the addition property. $$x-5=-8$$

Short Answer

Expert verified
The solution is \( x = -3 \).

Step by step solution

01

Understand the Addition Property

The addition property of equality states that you can add the same number to both sides of an equation without changing the equation. This is useful when we want to isolate the variable on one side of the equation.
02

Identify What to Add

We need to eliminate the -5 from the left side to isolate the variable \( x \). Therefore, we should add 5 to both sides of the equation.
03

Add to Both Sides

Add 5 to both sides of the equation: \( x - 5 + 5 = -8 + 5 \). This simplifies to \( x = -3 \).
04

Verify the Solution

Substitute \( x = -3 \) back into the original equation to check: \(-3 - 5 = -8 \). Since this statement is true, \( x = -3 \) is the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Linear Equations
Solving linear equations involves finding the value of the variable that makes the equation true. Linear equations typically look like a statement where two expressions are equal, such as \( x - 5 = -8 \).

To solve linear equations, we use basic arithmetic operations like addition, subtraction, multiplication, and division to isolate the variable on one side of the equation. The goal is to transform the equation until the variable is by itself, which allows us to see its value directly.

In our example, the equation involves subtraction. We aim to cancel out this subtraction to isolate \( x \), which leads us into the next concept: isolating variables.
Isolating Variables
The key to solving equations is isolating the variable, which means getting the variable alone on one side of the equation.

In the equation \( x - 5 = -8 \), the variable \( x \) is not yet isolated because it is being subtracted by 5. Using the addition property of equality, we can add the same number to both sides of the equation to counteract the subtraction.

Here, we add 5 to both sides to get rid of the \(-5\):
  • Add \(5\) to both sides: \( x - 5 + 5 = -8 + 5 \).
  • This simplifies to \( x = -3 \).

Thanks to the addition property, the 5 added to each side maintains the equality of the expression, allowing us to solve for \( x \) successfully.
Verifying Solutions
Once the variable has been isolated and a solution has been found, it is crucial to verify that this solution satisfies the original equation. Verification acts as a check to ensure no errors were made in the solving process.

To verify, substitute the value of the variable back into the original equation and perform any necessary calculations to confirm the statement is true.

In our example:
  • Substitute \( x = -3 \) into the original equation: \(-3 - 5\).
  • This simplifies to \(-8\), which matches the right side of the equation \(-8\).

The original statement \(-3 - 5 = -8\) holds true, confirming that \( x = -3 \) is indeed the correct solution. Verification is a powerful step to ensure accuracy and understanding of the solution process.

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Most popular questions from this chapter

Solve the equations by using the multiplication property. $$\frac{x}{3}=9$$

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