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To begin solving an equation like \( x + \frac{1}{3} = -\frac{1}{3} \), you'll want to isolate the variable, which in this case is \( x \). Isolating the variable means getting \( x \) alone on one side of the equation. This is essential because it allows us to directly determine its value, making solving the equation possible.
The process involves manipulating the equation to cancel out any other numbers or terms attached to \( x \). For our example, we subtract \( \frac{1}{3} \) from both sides of the equation. Why both sides? Because whatever we do to one side, we must do to the other to maintain the equality. This leads us to:

• Perform the operation: \( x + \frac{1}{3} - \frac{1}{3} = -\frac{1}{3} - \frac{1}{3} \)

The left side simplifies to \( x \), successfully isolating the variable, and the right side becomes \( -\frac{2}{3} \). This step sets the stage for finding the value of \( x \).

After isolating \( x \), the next step is to simplify the equation further. Simplifying an equation involves performing arithmetic operations to reduce it to its simplest form. In our case, this means dealing with the expression on the right side: \( -\frac{1}{3} - \frac{1}{3} \).
To simplify, you add the fractions together since they have a common denominator of 3. Always combine like terms where possible:

• Combine the numerators: \( -1 - 1 = -2 \)
• Keep the denominator the same: \( 3 \)

This leaves us with \( -\frac{2}{3} \). Now, the equation reads \( x = -\frac{2}{3} \), which is the simplest form of our equation and gives us the solution for \( x \). Simplifying like this helps in keeping the solution clear and easily understandable.

Once you have a solution, such as \( x = -\frac{2}{3} \), it’s important to check your work to ensure accuracy. Checking the solution involves substituting this value back into the original equation and verifying that both sides are equal. This step helps confirm that no mistakes were made during isolation or simplification.
Here's how you can check:

• Substitute \( x = -\frac{2}{3} \) into the original equation:
  \( -\frac{2}{3} + \frac{1}{3} = -\frac{1}{3} \)
• Perform the addition on the left:
  \( -\frac{2}{3} + \frac{1}{3} = -\frac{1}{3} \).

Both sides equal \( -\frac{1}{3} \), confirming that \( x = -\frac{2}{3} \) is indeed the correct solution. Always double-check to catch any potential errors along the way.