1. **State the problem:** We are given two linear equations: $$y = \frac{1}{2}x - 5$$ and $$y = -x + 4$$ We want to analyze these lines, find their intersection point, and understand their behavior. 2. **Recall the 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: $$\frac{1}{2}x - 5 = -x + 4$$ 3. **Solve for $x$:** $$\frac{1}{2}x - 5 = -x + 4$$ Add $x$ to both sides: $$\frac{1}{2}x + x - 5 = 4$$ Combine like terms: $$\frac{1}{2}x + \frac{2}{2}x - 5 = 4$$ $$\frac{3}{2}x - 5 = 4$$ Add 5 to both sides: $$\frac{3}{2}x = 9$$ Divide both sides by $\frac{3}{2}$: $$x = 9 \div \frac{3}{2} = 9 \times \frac{2}{3}$$ $$x = \cancel{9} \times \frac{2}{\cancel{3}} = 3 \times 2 = 6$$ 4. **Find $y$ by substituting $x=6$ into one of the equations:** Using $y = -x + 4$: $$y = -6 + 4 = -2$$ 5. **Interpretation:** The two lines intersect at the point $(6, -2)$. 6. **Summary:** - The first line has slope $\frac{1}{2}$ and y-intercept $-5$. - The second line has slope $-1$ and y-intercept $4$. - They intersect at $(6, -2)$. This means the lines cross at that point on the Cartesian plane.