Question: All three points displayed are on the line. Find an equation relating $x$ and $y$. Line graph in the top-right shows a straight increasing line passing through the points $(3,3)$, $(x,y)$, and $(6,9)$; position_hint \in \{top-right\}.
1. **State the problem:** We are given three points on a straight line: $(3,3)$, $(x,y)$, and $(6,9)$. We need to find an equation relating $x$ and $y$.
2. **Recall the formula for the equation of a line:** The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Since all three points lie on the same line, the slope between $(3,3)$ and $(x,y)$ must be equal to the slope between $(x,y)$ and $(6,9)$.
3. **Calculate the slope between $(3,3)$ and $(x,y)$:**
$$m_1 = \frac{y - 3}{x - 3}$$
4. **Calculate the slope between $(x,y)$ and $(6,9)$:**
$$m_2 = \frac{9 - y}{6 - x}$$
5. **Set the slopes equal:**
$$\frac{y - 3}{x - 3} = \frac{9 - y}{6 - x}$$
6. **Cross multiply to solve for $y$ in terms of $x$:**
$$(y - 3)(6 - x) = (9 - y)(x - 3)$$
7. **Expand both sides:**
$$6y - yx - 18 + 3x = 9x - 27 - yx + 3y$$
8. **Simplify and collect like terms:**
Left side: $6y - yx - 18 + 3x$
Right side: $9x - 27 - yx + 3y$
Bring all terms to one side:
$$6y - yx - 18 + 3x - 9x + 27 + yx - 3y = 0$$
Notice $-yx$ and $+yx$ cancel out:
$$6y - 18 + 3x - 9x + 27 - 3y = 0$$
Simplify:
$$ (6y - 3y) + (3x - 9x) + (-18 + 27) = 0$$
$$3y - 6x + 9 = 0$$
9. **Rewrite the equation:**
$$3y = 6x - 9$$
Divide both sides by 3:
$$y = \cancel{\frac{3}{3}} y = \frac{6x}{3} - \frac{9}{3}$$
$$y = 2x - 3$$
10. **Final answer:** The equation relating $x$ and $y$ for the points on the line is:
$$\boxed{y = 2x - 3}$$