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The distributive property allows us to simplify expressions by distributing a factor to each term within parentheses. For the exercise \( -\frac{4}{3}(y-12) - \frac{1}{6}y \), we first apply the distributive property to \( -\frac{4}{3} \) across the terms inside the parentheses.

This means multiplying \( -\frac{4}{3} \) with both \( y \) and \( -12 \):

• \( -\frac{4}{3} \cdot y = -\frac{4}{3}y \)
• \( -\frac{4}{3} \cdot (-12) = 16 \) (because multiplying two negatives gives a positive result)

By distributing the factor, we converted the expression to:

\( -\frac{4}{3}y + 16 \).
This step simplifies dealing with individual terms later on!

After distributing, we need to combine like terms. Like terms have the same variable raised to the same power. In our simplified expression, the like terms are those containing \( y \).

The given expression is:

\( -\frac{4}{3}y + 16 - \frac{1}{6}y \).

To combine like terms involving \( y \), we need a common denominator for the fractions. Here, the common denominator between 3 and 6 is 6.

So, \( -\frac{4}{3}y \) converts to \( -\frac{8}{6}y \). Now we have:

• \( -\frac{8}{6}y \)
• \( -\frac{1}{6}y \)

Combining these gives us:
\( -\frac{8}{6}y - \frac{1}{6}y = -\frac{9}{6}y \).
This simplifies further to:
\( -\frac{3}{2}y \).
Now, our expression reads:

\( -\frac{3}{2}y + 16 \).
Combine any remaining constants or similar terms to finalize the simplification.

Fraction arithmetic is crucial when simplifying algebraic expressions, especially when dealing with rational coefficients. It involves adding, subtracting, multiplying, and dividing fractions.

To simplify fractions, always find a common denominator when adding or subtracting. In our example, we needed to combine \( -\frac{4}{3}y \) and \( -\frac{1}{6}y \), so we found a common denominator of 6:

• \( -\frac{4}{3}y \) is converted to \( -\frac{8}{6}y \) by multiplying the numerator and the denominator by 2.

Properly combining fractions minimizes confusion and errors.

Additionally, simplifying fractions involves reducing them to their lowest terms. Finally, simplifying our expression reduces\( -\frac{9}{6}y \) to \( -\frac{3}{2}y \).

Working consistently with fraction arithmetic, especially with common denominators, makes tackling more complex expressions easier.