1. **State the problem:** We need to find the equation of a mystery line given several clues. 2. **Analyze Clue #1:** The mystery line is perpendicular to the line $$y = -\frac{2}{3}(x + 2) + 3$$. - The slope of the given line is $$m_1 = -\frac{2}{3}$$. - The slope of a line perpendicular to this is the negative reciprocal: $$m_2 = \frac{3}{2}$$. 3. **Analyze Clue #5:** The mystery line has a positive slope. - Our perpendicular slope $$\frac{3}{2}$$ is positive, so this matches. 4. **Analyze Clue #4:** Point A lies on the mystery line and also on the line $$y = 3$$. - So the y-coordinate of point A is 3. 5. **Analyze Clue #2:** The x-coordinate of point A is five less than its y-coordinate. - Since $$y_A = 3$$, then $$x_A = 3 - 5 = -2$$. - So point A is $$(-2, 3)$$. 6. **Write the equation of the mystery line:** - Using point-slope form with slope $$m = \frac{3}{2}$$ and point $$(-2, 3)$$: $$ y - 3 = \frac{3}{2}(x - (-2)) $$ $$ y - 3 = \frac{3}{2}(x + 2) $$ 7. **Simplify the equation:** $$ y = \frac{3}{2}x + \frac{3}{2} \times 2 + 3 = \frac{3}{2}x + 3 + 3 = \frac{3}{2}x + 6 $$ 8. **Analyze Clue #3:** The line will never pass through the fourth quadrant. - The fourth quadrant has $$x > 0$$ and $$y < 0$$. - Check if $$y = \frac{3}{2}x + 6$$ can be negative when $$x > 0$$. - For $$x > 0$$, $$y = \frac{3}{2}x + 6 > 6 > 0$$, so $$y$$ is always positive. - Therefore, the line never passes through the fourth quadrant, confirming the solution. **Final answer:** $$ y = \frac{3}{2}x + 6 $$