Question: Which of the following graphs represents the equation $6x + 3y = 18$?
Graph A: an increasing line crossing the y-axis at 6 and the x-axis at -3, position_hint \in \{top-left\}.
Graph B: a decreasing line crossing the y-axis at 6 and the x-axis at 4, position_hint \in \{top-right\}.
Graph C: a decreasing line crossing the y-axis at 4 and the x-axis at 6, position_hint \in \{bottom-left\}.
Graph D: a decreasing line crossing the y-axis at 6 and the x-axis at 18, position_hint \in \{bottom-right\}.
1. **State the problem:** We need to identify which graph corresponds to the equation $$6x + 3y = 18$$.
2. **Rewrite the equation in slope-intercept form:**
$$6x + 3y = 18$$
Subtract $$6x$$ from both sides:
$$3y = 18 - 6x$$
Divide both sides by $$3$$:
$$y = \frac{18}{3} - \frac{6x}{3}$$
Simplify:
$$y = 6 - 2x$$
3. **Interpret the slope-intercept form:**
- The slope $$m = -2$$ (negative, so the line is decreasing).
- The y-intercept $$b = 6$$ (the line crosses the y-axis at $$6$$).
4. **Find the x-intercept:** Set $$y=0$$:
$$0 = 6 - 2x$$
Add $$2x$$ to both sides:
$$2x = 6$$
Divide both sides by $$2$$:
$$x = \frac{6}{2} = 3$$
5. **Summary:**
- The line is decreasing (negative slope).
- Crosses y-axis at $$6$$.
- Crosses x-axis at $$3$$.
6. **Compare with given graphs:**
- Graph A: increasing line, y-intercept 6, x-intercept -3 (does not match slope or x-intercept).
- Graph B: decreasing line, y-intercept 6, x-intercept 4 (x-intercept incorrect).
- Graph C: decreasing line, y-intercept 4, x-intercept 6 (y-intercept incorrect).
- Graph D: decreasing line, y-intercept 6, x-intercept 18 (x-intercept incorrect).
None of the given graphs exactly match the x-intercept of $$3$$, but the closest is Graph B with x-intercept 4 and correct y-intercept and slope sign.
**Final answer:** None of the graphs perfectly match the equation $$6x + 3y = 18$$, but Graph B is the closest with a decreasing line and y-intercept at 6.
If the x-intercept in Graph B is a typo and should be 3, then Graph B is correct.