1. **State the problem:** We are given two linear equations: $$y = \frac{7}{2}x - 3$$ and $$y = -\frac{3}{2}x + 7$$ We want to analyze these lines, find their intersection point, and understand their behavior. 2. **Formula and rules:** To find the intersection of two lines, set their right-hand sides equal because at the intersection point, both $y$ values are the same: $$\frac{7}{2}x - 3 = -\frac{3}{2}x + 7$$ 3. **Solve for $x$:** Add $\frac{3}{2}x$ to both sides: $$\frac{7}{2}x + \frac{3}{2}x - 3 = 7$$ Combine like terms: $$\left(\frac{7}{2} + \frac{3}{2}\right)x - 3 = 7$$ $$\frac{10}{2}x - 3 = 7$$ $$5x - 3 = 7$$ Add 3 to both sides: $$5x - \cancel{3} + \cancel{3} = 7 + 3$$ $$5x = 10$$ Divide both sides by 5: $$\frac{\cancel{5}x}{\cancel{5}} = \frac{10}{5}$$ $$x = 2$$ 4. **Find $y$ coordinate:** Substitute $x=2$ into one of the original equations, for example: $$y = \frac{7}{2} \times 2 - 3 = 7 - 3 = 4$$ 5. **Interpretation:** The two lines intersect at the point $(2,4)$. 6. **Summary:** - The first line has slope $\frac{7}{2}$ and y-intercept $-3$. - The second line has slope $-\frac{3}{2}$ and y-intercept $7$. - They intersect at $(2,4)$. This means the lines cross at this point on the coordinate plane.