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When dealing with equations involving fractions, it's crucial to have a common ground to work from. Here, the concept of a common denominator becomes very handy. A common denominator is a shared multiple of the denominators of different fractions in an equation.
To find a common denominator, look at the denominators of the fractions involved. In this exercise, the fractions are \(\frac{x}{3}\) and \(\frac{x}{2}\). The respective denominators are 3 and 2. The smallest common multiple of these two numbers is 6.
By identifying a common denominator, you can rewrite each fraction so that they have the same base, making it easier to perform operations like addition or subtraction on them. Multiply each term by this common denominator to eliminate the fractions, simplifying your equation to something you can work with easily.

Simplifying an equation means changing it into a more straightforward form without altering its value. This involves performing operations to make the equation easier to solve.
Once you have a common denominator, multiply every term by that number to eliminate the fractions. In our example, multiplying the entire equation by 6 turns the original equation \(\frac{x}{3} + \frac{x}{2} = \frac{5}{6}\) into \(2x + 3x = 5\).
After multiplying, combine like terms. Like terms are terms that contain the same variable raised to the same power. In this case, \(2x\) and \(3x\) can be combined to give \(5x\).
It's always crucial to simplify equations as much as possible before finding solutions. This not only helps in making your calculations easier but also minimizes room for errors along the way.

Once you've found a solution to an equation, it's essential to verify that it's correct. This involves substituting your solution back into the original equation to see if both sides of the equation remain equal.
In this exercise, after solving for \(x\) and finding \(x = 1\), substitute \(1\) back into the original equation \(\frac{x}{3} + \frac{x}{2} = \frac{5}{6}\). Calculate each side: \(\frac{1}{3} + \frac{1}{2}\) simplifies to \(\frac{2}{6} + \frac{3}{6} = \frac{5}{6}\), confirming that the left side equals the right side.
If both sides of the equation match, it means the solution is correct. Checking solutions is a critical step to ensure accuracy, especially when dealing with more complex problems where errors are easily overlooked.