91Ó°ÊÓ

In algebra, isolating the variable means getting the unknown variable (in this case, \( t \)) alone on one side of the equation. This usually involves performing algebraic operations like addition, subtraction, multiplication, or division to both sides of the equation. Always ensure you perform the same operation on both sides to keep the equation balanced.

For example, in the equation:
\( t + \frac{1}{3} = \frac{1}{4} \),
our goal is to isolate \( t \). We start by eliminating \( \frac{1}{3} \) from the left side:
\( t + \frac{1}{3} - \frac{1}{3} = \frac{1}{4} - \frac{1}{3} \).
Notice how both sides of the equation have the same subtraction to keep them equal, leading to:
\( t = \frac{1}{4} - \frac{1}{3} \).

Once the variable is isolated, you can move on to other steps, such as simplifying or solving for the variable.

To perform operations on fractions, they must have the same denominator. This common denominator is usually the Least Common Denominator (LCD), which is the smallest number that both denominators divide into evenly.

Consider the fractions in our problem:
\( \frac{1}{4} \) and \( \frac{1}{3} \).
The denominators are 4 and 3. The LCD of 4 and 3 is 12, as it’s the smallest number divisible by both.

Convert the fractions to have the denominator of 12:
\[ \frac{1}{4} \rightarrow \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \]

\[ \frac{1}{3} \rightarrow \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \]

Now both fractions have the common denominator of 12, making them easier to work with.

Once your fractions have a common denominator, you can subtract them by simply dealing with the numerators while keeping the denominator the same.

In our example, converting the fractions to \( \frac{3}{12} \) and \( \frac{4}{12} \), we then subtract the numerators:
\( t = \frac{3}{12} - \frac{4}{12} \).
This results in:
\( t = \frac{3 - 4}{12} = \frac{-1}{12} \).

Remember, the denominators stay the same, and we only subtract the numerators to get the result. Don't forget, if the numerator ends up negative, the fraction is negative as well. Hence, our solution is \( t = \frac{-1}{12} \).