91Ó°ÊÓ

When solving linear equations, a key step is isolating the variable. This means to get the variable (like x) alone on one side of the equation. Let's look at how to isolate the variable in the equation: \ x + \frac{2}{3} = -\frac{1}{6} \The goal is to have x by itself on the left side. To do this, we need to move the fraction \frac{2}{3} to the right side. We can achieve this by subtracting \frac{2}{3} from both sides: \[ x + \frac{2}{3} - \frac{2}{3} = -\frac{1}{6} - \frac{2}{3} \] This simplifies to: \[ x = -\frac{1}{6} - \frac{2}{3} \] Now, x is isolated, meaning we’re one step closer to solving the equation.

Once we isolate the variable, the next step is to simplify the equation by combining any fractions that are left. From our previous step, we have: \[ x = -\frac{1}{6} - \frac{2}{3} \]To combine these fractions, they need to have a common denominator. Here, the common denominator is 6. The fraction \frac{2}{3} can be converted to \frac{4}{6}: \[ x = -\frac{1}{6} - \frac{4}{6} \]Now, combine the fractions by subtracting them: \[ x = -\frac{5}{6} \]Combining fractions makes the equation simpler and helps us find the value of the isolated variable.

After finding the solution to the equation, the final step is to verify that our solution is correct. This involves substituting the value of x back into the original equation to see if both sides are equal. Let's verify our solution, where we found x to be -\frac{5}{6}. Substitute this value back into the original equation: \[ x + \frac{2}{3} = -\frac{1}{6} \]Replace x with -\frac{5}{6}: \[ -\frac{5}{6} + \frac{2}{3} = -\frac{1}{6} \]Again, convert \frac{2}{3} to \frac{4}{6}: \[ -\frac{5}{6} + \frac{4}{6} = -\frac{1}{6} \]Since both sides of the equation are equal, we have verified that our solution x = -\frac{5}{6} is correct. Verifying solutions ensures accuracy and eliminates any errors in the solving process.