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Algebraic fractions are similar to regular fractions but contain variables in their numerators, denominators, or both. Understanding how to manipulate these expressions is essential for solving equations involving fractions. The key in dealing with algebraic fractions is to clear the fractions by finding a common denominator, which allows you to work with whole numbers or 'cleaner' algebraic expressions.

For example, in the given problem, \(\frac{x}{4}=2+\frac{x-3}{3}\), we have \(x\) divided by 4 and \(x-3\) divided by 3. To rid the equation of fractions, we multiply every term by the least common multiple (LCM) of the denominators, in this case, 12. This step is crucial as it simplifies the equation and paves the way for easier manipulation of the terms.

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers. In the context of algebra, finding the LCM of the denominators in algebraic fractions helps you combine these fractions by creating a common denominator.

In our exercise, we needed to find the LCM of 4 and 3 to eliminate the algebraic fractions. Since 4 and 3 are relatively prime (they have no common factors other than 1), their LCM is simply their product, 12. This means that if we multiply both sides of the equation by 12, the denominators will cancel out, leaving us with an equation free of fractions. Understanding how to find the LCM is thus a critical step in the process of solving equations with algebraic fractions.

Combining like terms involves simplifying algebraic expressions by adding or subtracting terms that have the same variables raised to the same power. It's an essential process that helps in reducing equations to their simplest form, making them easier to solve.

Once we've eliminated the algebraic fractions in our original problem, we obtained the equation \(3x=24+4(x-3)\). The next step is to distribute the 4 into the parentheses and then combine terms that contain the variable \(x\), as well as any constant terms. So the expression \(4(x-3)\) becomes \(4x - 12\), and the equation simplifies when we combine the \(x\) terms. This combined expression reveals a simpler relation which we can then manipulate to solve for the unknown variable \(x\).

To isolate the variable means to get the variable on one side of the equation and all the other terms on the opposite side, which sets the stage for finding the solution of the equation. The goal is to have the variable by itself, expressed as equal to a number or simpler expression.

In the final steps of solving the given problem, we isolated \(x\) by moving all terms containing \(x\) to one side of the equation, resulting in \(3x - 4x = 12\). This reduces to \(\-x = 12\), allowing us to easily find the solution by multiplying both sides of the equation by -1, giving us \(x = -12\). Isolating the variable is often the penultimate step in solving linear equations, with the final step being to ensure the variable is positive for a straightforward solution.