1. The problem is to analyze the system of linear equations from set 7: $$y = -\frac{1}{2}x - 2$$ $$y = -\frac{3}{2}x + 2$$ 2. We want to find the point of intersection of these two lines, if any. 3. Set the right-hand sides equal since at the intersection point both $y$ values are the same: $$-\frac{1}{2}x - 2 = -\frac{3}{2}x + 2$$ 4. Add $\frac{3}{2}x$ to both sides: $$-\frac{1}{2}x + \frac{3}{2}x - 2 = 2$$ 5. Simplify the left side: $$\left(-\frac{1}{2} + \frac{3}{2}\right)x - 2 = 2$$ $$\frac{2}{2}x - 2 = 2$$ $$x - 2 = 2$$ 6. Add 2 to both sides: $$x - \cancel{2} + \cancel{2} = 2 + 2$$ $$x = 4$$ 7. Substitute $x=4$ into the first equation to find $y$: $$y = -\frac{1}{2} \times 4 - 2 = -2 - 2 = -4$$ 8. The point of intersection is: $$(4, -4)$$ This means the two lines cross at the point where $x=4$ and $y=-4$. "slug": "line intersection", "subject": "algebra", "desmos": {"latex": "y=-\frac{1}{2}x-2, y=-\frac{3}{2}x+2", "features": {"intercepts": true, "extrema": false}}, "q_count": 2