91Ó°ÊÓ

Isolating a variable is a fundamental technique in solving linear equations. It means getting the variable by itself on one side of the equation. In our exercise, the variable is y. Start with the equation:\[ y + \frac{1}{3} = \frac{4}{3} \]To isolate y, we need to remove \( \frac{1}{3} \) from the left side. We do this by performing the inverse operation, which is subtraction.Subtract \( \frac{1}{3} \) from both sides of the equation:
\[ y + \frac{1}{3} - \frac{1}{3} = \frac{4}{3} - \frac{1}{3} \]This leaves us with:
\[ y = \frac{4}{3} - \frac{1}{3} \]By isolating the variable, we effectively simplify our work and can now find the value of y.

Simplifying fractions is the process of making a fraction simpler by combining or reducing its components. In our exercise, after isolating the variable y, we have the equation:\[ y = \frac{4}{3} - \frac{1}{3} \]To solve this, we need to subtract these two fractions. Here's how to do it step-by-step:
1. Ensure the fractions have a common denominator, which they already do (3).
2. Subtract the numerators:
\[ \frac{4}{3} - \frac{1}{3} = \frac{4 - 1}{3} = \frac{3}{3} \]3. Simplify the resulting fraction:
\[ \frac{3}{3} = 1 \]By simplifying fractions, we convert complex fractions into simpler forms that are easier to interpret and solve.

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. In our given exercise:
\[ y + \frac{1}{3} = \frac{4}{3} \]This is a linear equation because it has the form y plus a constant equals another constant. We solve linear equations to find the value of the variable that makes the equation true. Here, by isolating y and simplifying the fractions, we found that:
\[ y = 1 \]Linear equations form the foundation of algebra and are crucial in understanding how to solve more complex mathematical problems. They are always solved by isolating the variable and performing arithmetic operations to simplify the equation.