solving equations using the additive inverse

Solving equations using the additive inverse involves the concept of "undoing" addition or subtraction in an equation to isolate the variable. The additive inverse of a number is simply the number that, when added to the original number, results in zero. For example, the additive inverse of \(5\) is \(-5\), and the additive inverse of \(-7\) is \(7\).

Here’s a step-by-step guide:

### Example Problem:
Solve the equation:  
\[
x + 5 = 12
\]

### Steps:

1. **Identify the term that needs to be eliminated**:  
   The goal is to isolate \(x\) by eliminating the \(+5\).

2. **Apply the additive inverse**:  
   The additive inverse of \(+5\) is \(-5\). Add \(-5\) to both sides of the equation to cancel out the \(+5\):
   \[
   x + 5 - 5 = 12 - 5
   \]
   
3. **Simplify both sides**:  
   On the left side, \(+5 - 5 = 0\), so:
   \[
   x = 7
   \]

4. **Solution**:  
   The solution is \(x = 7\).

### Another Example:
Solve the equation:  
\[
y - 8 = 3
\]

1. **Identify the term that needs to be eliminated**:  
   The goal is to eliminate the \(-8\).

2. **Apply the additive inverse**:  
   The additive inverse of \(-8\) is \(+8\). Add \(8\) to both sides:
   \[
   y - 8 + 8 = 3 + 8
   \]

3. **Simplify both sides**:  
   \[
   y = 11
   \]

4. **Solution**:  
   The solution is \(y = 11\).

### Key Idea:
To solve an equation using the additive inverse, always add or subtract the same amount on both sides of the equation to cancel out the term that involves the variable.

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Do you want more details or have any questions?

Here are 5 related questions:
1. How would you solve \( z - 6 = 9 \) using the additive inverse?
2. Can you solve an equation like \( 3x + 7 = 16 \) using additive inverse first, then division?
3. How can the additive inverse be used in solving equations with decimals, like \( x + 2.5 = 5.8 \)?
4. How do you apply the additive inverse to solve inequalities?
5. Can the additive inverse be used in solving systems of equations?

**Tip**: Always remember that whatever operation you apply to one side of the equation must also be applied to the other side to keep the equation balanced.

Learn how to solve equations using the additive inverse, a core concept in algebra. This guide provides examples of isolating variables by applying the additive inverse property. Suitable for students in Grades 6-8.

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