91Ó°ÊÓ

Understanding how to subtract fractions is crucial in solving equations involving fractions. It can be a bit tricky at first, but it's all about following a certain set of rules.

When you subtract fractions, they must have the same denominator. The denominator is the bottom part of the fraction and represents how many equal parts the whole is divided into. If the denominators are not the same, you can't perform the subtraction directly. Let's look at our given problem as an example:

We need to subtract these fractions: \[ \frac{2}{3} - \frac{9}{10} \]

Since the denominators (3 and 10) are different, we need to find a common denominator before we can subtract the numerators (top parts of the fractions).

A common denominator is a common multiple of the denominators of two or more fractions. In this case, the denominators are 3 and 10.

To find a common denominator, look for the smallest number that both denominators can divide into without leaving a remainder. The smallest common multiple of 3 and 10 is 30.

We convert each fraction to an equivalent fraction with the same denominator.
Convert \[ \frac{2}{3} \] to a fraction with denominator 30 by multiplying the numerator and denominator by 10:
\[ \frac{2}{3} \times \frac{10}{10} = \frac{20}{30} \]

Now do the same for \[ \frac{9}{10} \]:
\[ \frac{9}{10} \times \frac{3}{3} = \frac{27}{30} \]

With both fractions having the same denominators, fraction subtraction becomes straightforward.

Isolating the variable is the main step to solve an equation containing a variable. Our goal is to get the variable, in this case, \(g\), by itself on one side of the equation.

The equation we are working with after translating the words is:
\[ \frac{9}{10} + g = \frac{2}{3} \]

To isolate \(g\), we need to remove \[ \frac{9}{10} \] from the left side of the equation. We do this by subtracting \[ \frac{9}{10} \] from both sides:
\[ g = \frac{2}{3} - \frac{9}{10} \]

Once we have found a common denominator and made the subtraction possible, we get:

\[ g = \frac{20}{30} - \frac{27}{30} = \frac{-7}{30} \]

This simple approach of isolating variables through fraction subtraction highlights the importance of understanding both concepts deeply.