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The addition principle is a key concept in solving equations. It states that if you add or subtract the same value from both sides of an equation, the equation remains balanced. This principle is crucial for isolating variables and solving for unknowns. Let's break it down with an example:

Given the equation:

$$ r + \frac{1}{3} = \frac{8}{3} $$

To isolate the variable \(r\), we use the addition principle by subtracting \( \frac{1}{3} \) from both sides:

$$ r + \frac{1}{3} - \frac{1}{3} = \frac{8}{3} - \frac{1}{3} $$

Notice how \( \frac{1}{3} \) is subtracted from both sides, maintaining the balance of the equation. This step helps us simplify the equation and isolate \(r\).

Key takeaways:

• Maintain the balance of the equation.
• Add or subtract the same value from both sides.

Isolating the variable means rearranging the equation so that the unknown variable is alone on one side of the equation. This step is essential in solving equations, as it makes it easier to identify the value of the unknown variable.

Let's continue with our equation:

$$ r + \frac{1}{3} = \frac{8}{3} $$

As mentioned earlier, to isolate \(r\), we subtract \( \frac{1}{3} \) from both sides:

$$ r + \frac{1}{3} - \frac{1}{3} = \frac{8}{3} - \frac{1}{3} $$

This simplifies to:

$$ r = \frac{7}{3} $$

Now, \(r\) is isolated and its value is clear. This step-by-step process of isolating the variable helps in solving more complex equations as well.

Key points:

• Isolate the unknown variable on one side.
• Use principles like addition or subtraction to achieve this.

Simplifying fractions involves making a fraction as simple as possible. This means reducing it to its lowest terms or combining fractions to simplify an equation.

In our equation:

$$ r + \frac{1}{3} = \frac{8}{3} $$

We isolated \(r\) by applying the addition principle:

$$ r = \frac{8}{3} - \frac{1}{3} $$

The next step is simplifying the right-hand side. Subtracting the fractions involves finding a common denominator. Since they already have the same denominator (3), we can directly subtract:

$$ \frac{8}{3} - \frac{1}{3} = \frac{8-1}{3} = \frac{7}{3} $$

Thus, we simplified the fractions to get \( \frac{7}{3} \).

Tips for simplifying fractions:

• Find a common denominator if the fractions are different.
• Combine or reduce fractions to their simplest form.