91Ó°ÊÓ

In algebra, one of the most important steps in solving an equation is isolating the variable. This means getting the variable (like 'x' in our equation) by itself on one side of the equation. To do this, you generally perform the same mathematical operation on both sides of the equation.
For example, in our problem, the equation given is \(\frac{2}{3}x = \frac{9}{2}\). This means we need to isolate 'x' so we can figure out what value it holds. To isolate 'x', you can multiply both sides of the equation by the reciprocal of \(\frac{2}{3}\). By doing so, we essentially cancel out the fraction next to 'x', leaving 'x' alone on one side of the equation.
Isolating variables is crucial because it transforms complex equations into simpler ones. Remember, whatever you do on one side of the equation, you must do the same to the other side.

A reciprocal is a number that, when multiplied by the original number, results in 1. For example, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). When you multiply \(\frac{2}{3}\) by \(\frac{3}{2}\), it equals 1 (\(\frac{2}{3} \times \frac{3}{2} = 1\)).
In our problem, we use the reciprocal to help isolate the variable. Specifically, we multiply both sides of the equation \(\frac{2}{3} x = \frac{9}{2}\) by \(\frac{3}{2}\).
Reciprocals are helpful because they enable us to get rid of fractions. When dealing with linear equations, they provide a quick way to isolate variables, simplifying the problem.

Multiplying fractions involves multiplying the numerators together and the denominators together. For example, to multiply \(\frac{2}{3}\) by \(\frac{3}{2}\), you multiply the numerators: 2 and 3 (resulting in 6), and the denominators: 3 and 2 (resulting in 6). The result is \(\frac{6}{6} = 1\).
When solving our exercise, we multiply \(\frac{3}{2}\) by both sides of the equation \(\frac{2}{3} x = \frac{9}{2}\). This step includes:

• Multiplying \(\frac{3}{2} \times \frac{2}{3} x\) to isolate 'x'.
• Multiplying \(\frac{3}{2} \times \frac{9}{2}\) to work on the other side of the equation, resulting in \(\frac{27}{4}\).

These steps hinge on knowing how to multiply fractions correctly, which is fundamental knowledge for working with linear equations.