91Ó°ÊÓ

When you encounter equations with fractions, one effective method for solving them is cross-multiplication. This strategy is particularly useful when two fractions are set equal to each other, forming a proportion. Here’s the essence of this process: multiply the numerator of the first fraction by the denominator of the second fraction and set this product equal to the product of the denominator of the first fraction and the numerator of the second fraction.

For instance, in the equation \(\frac{12}{x} = \frac{3}{2}\), cross-multiplication entails multiplying 12 by 2, and then 'x' by 3, resulting in the equation '24 = 3x'. This transforms the proportion into a much simpler linear equation that is easier to solve. The benefit of cross-multiplication is that it removes fractions from the equation, simplifying the path to isolating the variable.

The technique of isolating variables is at the heart of solving equations. This involves manipulating the equation in such a way that the variable we want to solve for, typically 'x', is on one side of the equation by itself.

After cross-multiplying, we got to the point where the equation looked like '24 = 3x'. To isolate 'x', we need to perform operations that will undo the multiplication of 'x' by 3. We achieve this by dividing both sides of the equation by 3. Mathematically, we express this step as \(x = \frac{24}{3}\). Through this operation, 'x' is isolated, and the equation becomes clearer and straightforward to solve. Remember that whatever operation you perform on one side of an equation, you must do the same to the other side to maintain balance.

Once we've successfully isolated the variable, the next step is to simplify the equation. Simplification involves performing arithmetic to reduce the equation to its most basic form, making it easy to identify the solution.

In our example, after isolating 'x' to one side, we have \(x = \frac{24}{3}\). Upon simplifying this fraction, we divide 24 by 3, giving us 'x = 8'. The simplification step is crucial because it delivers the answer in the clearest possible manner. When simplifying, always break down numbers to their simplest form, cancel out common factors if possible, and ensure that the expression on both sides of the equation is as straightforward as it can be to interpret the solution easily.