91Ó°ÊÓ

When solving algebraic equations, isolating the variable is a crucial first step. In our exercise, we start with the equationewline ewline \(\frac{2}{3}p = \frac{1}{2} p + \frac{1}{3}\).ewline ewline To isolate the variable term, we must move all terms containing the variable to one side of the equation. This helps simplify the equation and focus on solving for the unknown. We achieve this by subtracting \(\frac{1}{2}p\) from both sides: ewline ewline \(\frac{2}{3}p - \frac{1}{2}p = \frac{1}{3}\).ewline ewline As you can see, isolating the variable involves basic arithmetic operations like addition, subtraction, multiplication, or division. This makes it easier to combine and compare like terms.

Once the variable terms are isolated, the next step is to combine them. In our example, after isolating terms, we have the expression:ewline ewline \(\frac{2}{3}p - \frac{1}{2}p = \frac{1}{3}\).ewline ewline To combine these terms, they must share a common denominator. In this case, the common denominator between 3 and 2 is 6. We convert the fractions to have this common denominator: ewline ewline \(\frac{4}{6}p - \frac{3}{6}p = \frac{1}{3}\).ewline ewline This step transforms our expression into:ewline ewline \(\frac{1}{6}p = \frac{1}{3}\).ewline ewline Combining like terms simplifies the equation further and prepares us for the final step of solving for the variable.

Finding a common denominator is an essential skill when working with fractions. It allows us to add or subtract fractions easily. In our problem, the goal was to combine like terms involving fractions:ewline ewline \(\frac{4}{6}p - \frac{3}{6}p = \frac{1}{3}\).ewline ewline The common denominator of 3 and 2 is 6. This step ensures that all fractions are expressed with the same base, making them straightforward to manipulate. For example, converting \(\frac{2}{3}\) and \(\frac{1}{2}\) to \(\frac{4}{6}\) and \(\frac{3}{6}\) respectively enables us to subtract directly. ewline ewline Remember, the common denominator must be a multiple of the original denominators. This step prepares the equation for simpler operations, especially when fractions are involved. It's crucial for precise calculations and achieving accurate results when solving equations.