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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.

For example, if you have an equation like: \( a = b \), and you add the same number, say \( c \), to both sides, your new equation will be \( a + c = b + c \). This property helps in maintaining balance on both sides of the equation.

In the given example of \( r + 8 = 12 \), you can think of 8 as an intruder that needs to be moved.

By subtracting 8 from both sides, you balance the equation and keep the equality intact. Understanding this concept is important because it is a building block for more complex algebraic manipulations.

Isolating the variable is a crucial step in solving equations. It involves moving all terms containing the variable to one side of the equation and everything else to the other.

In the given equation \( r+8=12 \), the variable \( r \) is mixed with a number 8. To isolate \( r \), you need to remove 8 from the left side. You do this by subtracting 8 from both sides of the equation: \( r+8-8=12-8 \). This approach helps in simplifying the equation so you can easily find the value of the variable.

When isolating variables, always remember to perform the same operation on both sides of the equation. This way, the equality stays true, no matter what operation you perform.

Simplifying equations is the step where you bring your equation to its simplest form. After isolating the variable, you need to ensure there are no more calculations to be done.

For the equation \( r+8-8=12-8 \), simplifying involves performing the subtraction \( 8-8 \) on the left side, which results in 0: \( r+0 \). On the right side, subtract 8 from 12 to get 4: \( 12-8 = 4 \).

Thus, the equation simplifies to: \( r = 4 \).

Simplification helps clearly identify the value of the variable. The goal is to make the equation as straightforward as possible.

Breaking down each step while simplifying ensures you don't miss any important detail and arrive at the correct answer.