Q. \( \frac{1}{2}x – \frac{1}{3}y + xy – \frac{6}{5}y^{2} \).

Problem

Simplify and analyze the expression:

\(\frac{1}{2}x – \frac{1}{3}y + xy – \frac{6}{5}y^2\)

Step 1: Identify each term

The expression is:

\(\frac{1}{2}x – \frac{1}{3}y + xy – \frac{6}{5}y^2\)

It has four terms:

\(\frac{1}{2}x\)

\(-\frac{1}{3}y\)

\(xy\)

\(-\frac{6}{5}y^2\)

The variables are \(x\) and \(y\). The expression includes linear terms, a product term, and a squared term.

Step 2: Check for like terms

Like terms have the same variables raised to the same powers.

Now compare the terms:

\(\frac{1}{2}x\) has only \(x\).

\(-\frac{1}{3}y\) has only \(y\).

\(xy\) has both \(x\) and \(y\).

\(-\frac{6}{5}y^2\) has \(y^2\).

Since none of these terms have the same variable part, there are no like terms to combine.

Step 3: Rewrite the expression in a clear order

A common order is to place the product term first, then the \(y^2\) term, then the linear terms:

\(xy – \frac{6}{5}y^2 + \frac{1}{2}x – \frac{1}{3}y\)

This is equivalent to the original expression. We only changed the order of the terms.

Step 4: Look for a common factor

The expression is:

\(xy – \frac{6}{5}y^2 + \frac{1}{2}x – \frac{1}{3}y\)

Not every term has the same variable factor.

The terms \(xy\), \(-\frac{6}{5}y^2\), and \(-\frac{1}{3}y\) contain \(y\), but \(\frac{1}{2}x\) does not.

The terms \(xy\) and \(\frac{1}{2}x\) contain \(x\), but the other terms do not.

So there is no single variable factor common to all four terms.

Step 5: Clear the fractions

The denominators are:

\(2\), \(3\), and \(5\)

The least common denominator is:

\(30\)

To rewrite the expression with one denominator, express every coefficient over \(30\).

First term:

\(\frac{1}{2}x = \frac{15}{30}x\)

Second term:

\(-\frac{1}{3}y = -\frac{10}{30}y\)

Third term:

\(xy = \frac{30}{30}xy\)

Fourth term:

\(-\frac{6}{5}y^2 = -\frac{36}{30}y^2\)

Now write the whole expression over \(30\):

\(\frac{15x – 10y + 30xy – 36y^2}{30}\)

Step 6: Check whether the numerator can factor

The numerator is:

\(15x – 10y + 30xy – 36y^2\)

Rewrite it in a clearer order:

\(30xy – 36y^2 + 15x – 10y\)

Now try factoring by grouping:

\((30xy – 36y^2) + (15x – 10y)\)

Factor the first group:

\(30xy – 36y^2 = 6y(5x – 6y)\)

Factor the second group:

\(15x – 10y = 5(3x – 2y)\)

The two groups do not share the same binomial factor.

So this grouping does not produce a simple factorization.

Step 7: Final simplified form

The original expression is already simplified because there are no like terms to combine.

A clean version is:

\(xy – \frac{6}{5}y^2 + \frac{1}{2}x – \frac{1}{3}y\)

An equivalent single-fraction form is:

\(\frac{30xy – 36y^2 + 15x – 10y}{30}\)

Final answer

\(\frac{1}{2}x – \frac{1}{3}y + xy – \frac{6}{5}y^2 = xy – \frac{6}{5}y^2 + \frac{1}{2}x – \frac{1}{3}y\)

Equivalently:

\(\frac{1}{2}x – \frac{1}{3}y + xy – \frac{6}{5}y^2 = \frac{30xy – 36y^2 + 15x – 10y}{30}\)