91Ó°ÊÓ

When solving equations, the addition principle allows us to maintain equality by adding or subtracting the same value from both sides of the equation. This becomes especially useful when we need to eliminate extra terms from one side of the equation.

Consider the equation: \( r + \frac{1}{3} = \frac{8}{3} \).
To isolate the variable \(r\), we can subtract \( \frac{1}{3} \) from both sides of the equation. This step ensures that we're treating both sides equally, preserving the balance of the equation. So, we get:
\[ r + \frac{1}{3} - \frac{1}{3} = \frac{8}{3} - \frac{1}{3} \]
After performing the subtraction, the equation simplifies to:
\[ r = \frac{7}{3} \]
Through this step, we have successfully isolated the variable on one side using the addition principle.

Isolating the variable is a crucial step in solving any linear equation. It involves getting the variable by itself on one side of the equation, allowing us to solve for its value. Let's look at our current example: \( r + \frac{1}{3} = \frac{8}{3} \).

We want to isolate \(r\), so we subtract \( \frac{1}{3} \) from both sides. This gives:
\[ r + \frac{1}{3} - \frac{1}{3} = \frac{8}{3} - \frac{1}{3} \]
Simplifying, we get:
\[ r = \frac{7}{3} \]
This method clears the way to easily identify the value of the variable.

By isolating the variable, we simplify the problem. It's easier to then substitute this value back into the original equation for verification.

Once we've found the solution, it's important to verify it by substituting the value back into the original equation. This ensures there were no mistakes in our calculations and that the solution is indeed correct. Let's check our solution for \( r = \frac{7}{3} \).

Original equation: \( r + \frac{1}{3} = \frac{8}{3} \)
Substitute \( r = \frac{7}{3} \):
\[ \frac{7}{3} + \frac{1}{3} = \frac{8}{3} \]
Simplifying the left-hand side, we get:
\[ \frac{7}{3} + \frac{1}{3} = \frac{8}{3} \]
since \( \frac{8}{3} = \frac{8}{3} \) is a true statement, we confirm that our solution \( r = \frac{7}{3} \) is correct.

Always remember to check your solutions to ensure accuracy and reinforce your understanding of the problem-solving process. This step helps catch any potential errors and confirms your solution is valid.