1. **State the problem:** We are given two linear equations: $$y = \frac{2}{3}x + 2$$ and $$y = 2x - 6$$ We want to analyze these lines, find their intersection point, and understand their slopes and intercepts. 2. **Recall the formula and rules:** - The slope-intercept form of a line is $$y = mx + b$$ where $$m$$ is the slope and $$b$$ is the y-intercept. - The intersection point of two lines is where their $$y$$ values are equal. 3. **Set the two equations equal to find the intersection:** $$\frac{2}{3}x + 2 = 2x - 6$$ 4. **Solve for $$x$$:** Subtract $$\frac{2}{3}x$$ from both sides: $$2 = 2x - 6 - \frac{2}{3}x$$ Rewrite the right side: $$2 = \left(2 - \frac{2}{3}\right)x - 6$$ Calculate the coefficient: $$2 - \frac{2}{3} = \frac{6}{3} - \frac{2}{3} = \frac{4}{3}$$ So: $$2 = \frac{4}{3}x - 6$$ Add 6 to both sides: $$2 + 6 = \frac{4}{3}x$$ $$8 = \frac{4}{3}x$$ 5. **Isolate $$x$$:** $$x = \frac{8}{\frac{4}{3}} = 8 \times \frac{3}{4}$$ Show cancellation: $$x = \frac{\cancel{8}}{\cancel{4}} \times \frac{3}{1} = 2 \times 3 = 6$$ 6. **Find $$y$$ by substituting $$x=6$$ into one of the equations:** Using $$y = 2x - 6$$: $$y = 2(6) - 6 = 12 - 6 = 6$$ 7. **Interpretation:** - The lines intersect at the point $$(6, 6)$$. - The first line has slope $$\frac{2}{3}$$ and y-intercept 2. - The second line has slope 2 and y-intercept -6. - Since the slopes are different, the lines are not parallel and intersect at exactly one point. **Final answer:** The two lines intersect at the point $$(6, 6)$$.