Factor 2x - 9xy.

Step-by-step solution

1. 1

   Compare the two terms factor by factor. Write each out:

   $2x=2\cdot x,\qquad9xy=9\cdot x\cdot y$

   The coefficients $2$ and $9$ are coprime, so no number comes out. The variable $y$ appears only in the second term, so it cannot come out either. The only shared factor is $x$.

2. 2

   Divide each term by $x$.

   $\frac{2x}{x}=2,\qquad\frac{-9xy}{x}=-9y$

   so the bracket holds $2-9y$, with the minus sign carried across:

   $2x-9xy=x(2-9y)$

3. 3

   Check by expanding. $x\cdot2-x\cdot9y=2x-9xy$ ✓, confirmed at all $400$ integer pairs with $-10\le x,y\le9$ ✓. Inside the bracket $2$ and $9y$ share nothing, so the factoring is complete.

4. 4

   Solve the associated equation. Setting the product to zero and using the zero-product property:

   $x(2-9y)=0\quad\Longrightarrow\quad x=0\ \text{ or }\ 2-9y=0$

   The second branch gives $9y=2$, so $y=\tfrac29$.

5. 5

   Interpret the solution set. Because there are two variables, the solutions form two whole lines in the $(x,y)$ plane: the vertical line $x=0$ (any $y$) and the horizontal line $y=\tfrac29$ (any $x$). Check a point on each: $x=0,y=5$ gives $0-0=0$ ✓; $x=7,y=\tfrac29$ gives $14-9\cdot7\cdot\tfrac29=14-14=0$ ✓.