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When solving equations with fractions, it is often useful to eliminate the fractions to make the equation easier to work with. One way to do this is by finding the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that all denominators can divide into without leaving a remainder.

For example, given the denominators 2, 3, 9, and 6, the LCM is 18.

• 2 divides into 18 exactly 9 times.
• 3 divides into 18 exactly 6 times.
• 9 divides into 18 exactly 2 times.
• 6 divides into 18 exactly 3 times.

Once the LCM is found, each term in the equation should be multiplied by this LCM. This will effectively eliminate the fractions and make the equation easier to solve.

Isolating the variable is a critical step in solving equations. The goal is to get the variable on one side of the equation and all the other terms on the opposite side.

To do this, we use various techniques of equation manipulation, such as addition, subtraction, multiplication, and division.

In our example: \(\frac{x}{2}+\frac{1}{3}=\frac{x}{9}+\frac{1}{6}\), after eliminating the fractions, we get: \(9x + 6 = 2x + 3\).

We can isolate \(x\) by subtracting \(2x\) from both sides, which gives: \(9x - 2x + 6 = 3\), simplifying to \(7x + 6 = 3\).

Then subtract 6 from both sides: \(7x = 3 - 6\), simplifying to \(7x = -3\). Finally, divide both sides by 7: \(x = \frac{-3}{7}\).

Simplifying fractions involves reducing them to their simplest form. This step is very important when working with equations that contain fractions because it makes the calculations easier and the numbers more manageable.

For example, to simplify \( \frac{18 \times \frac{1}{3}}{1} = 6\), we recognize that 18 is divisible by 3, which gives 6 as a result. Similarly, \( 18 \times \frac{x}{9} = 2x\) is simplified by dividing 18 by 9, resulting in 2x.

• Always check each multiplication to ensure that the fraction is simplified correctly.
• Simplifying helps in preventing calculation errors.
• It also speeds up the process of solving the equation.

Equation manipulation involves using various algebraic techniques to simplify and solve equations. Key techniques include adding, subtracting, multiplying, and dividing both sides of the equation by the same number.

When fractions are present, first eliminate them using the LCM, then perform the algebraic manipulations.

• For example, in the equation \(9x + 6 = 2x + 3\), we subtracted \(2x\) from both sides to get \(9x - 2x + 6 = 3\).
• Then simplify: \(7x + 6 = 3\).
• Next, subtract 6: \(7x = -3\).
• Finally, divide by 7 to solve for x: \(x = \frac{-3}{7}\).

Each of these steps is part of manipulating the equation to isolate the variable and find its value. Following these steps in order ensures the equation remains balanced and the solution is correct.