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The Addition Property of Equality is a fundamental concept in algebra. It states that if you add the same number to both sides of an equation, the equality is still true. This property is especially useful when you want to isolate a variable.
Imagine you have an equation like:
\[ x - 5 = 10 \]
To isolate \( x \), you can add 5 to both sides of the equation:
\[ x - 5 + 5 = 10 + 5 \]
Which simplifies to:
\[ x = 15 \]
By applying the Addition Property of Equality correctly, we can easily move terms across the equal sign and maintain a balanced equation.

The Subtraction Property of Equality is another key tool in algebra. It states that if you subtract the same number from both sides of an equation, the equality holds true. This property helps in isolating a variable for solving linear equations.
Let's say you have the equation:
\[ y + 8 = 15 \]
To isolate \( y \), you'll subtract 8 from both sides:
\[ y + 8 - 8 = 15 - 8 \]
Simplifying this, you get:
\[ y = 7 \]
Using the Subtraction Property of Equality allows you to remove terms from one side of the equation, making it simpler to find the value of the variable.

In our original problem, we used this property to isolate the variable \( m \):
\[ m + 7.9 = 11.6 \]
By subtracting 7.9 from both sides:
\[ m + 7.9 - 7.9 = 11.6 - 7.9 \]
We obtained:
\[ m = 3.7 \] This makes it easier to see how subtraction helps in solving equations.

Solving linear equations involves finding the value of the variable that makes the equation true. Here’s a step-by-step method:
1. **Understand the Equation**: Identify the given equation and determine what you need to find.
2. **Isolate the Variable**: Use properties like addition or subtraction to get the variable alone on one side of the equation.
3. **Simplify**: Perform the operations needed to solve for the variable.
4. **Verify the Solution**: Substitute your solution back into the original equation to ensure both sides are equal.

Let's go through a practical example using these steps:
Given the equation \( m + 7.9 = 11.6 \), you need to isolate \( m \).
By using the Subtraction Property of Equality, subtract 7.9 from both sides:
\[ m + 7.9 - 7.9 = 11.6 - 7.9 \]
Simplify to get:
\[ m = 3.7 \]
To verify, substitute \( m = 3.7 \) back into the original equation:
\[ 3.7 + 7.9 = 11.6 \]
Both sides of the equation are equal, confirming that the solution is correct.
By following these steps, you'll be able to solve linear equations effectively and with confidence.