Line Intersection B6315B

1. **State the problem:** We are given two linear equations: $$y = 2x + 1$$ and $$y = 3x + 5$$. We need to find the point where these two lines intersect. 2. **Set the equations equal:** Since both expressions equal $$y$$, set them equal to each other to find $$x$$: $$2x + 1 = 3x + 5$$ 3. **Solve for $$x$$:** Subtract $$2x$$ from both sides: $$\cancel{2x} + 1 = 3x + 5 - \cancel{2x}$$ which simplifies to: $$1 = x + 5$$ Subtract $$5$$ from both sides: $$1 - 5 = x + 5 - 5$$ $$-4 = x$$ 4. **Find $$y$$:** Substitute $$x = -4$$ into one of the original equations, for example, $$y = 2x + 1$$: $$y = 2(-4) + 1 = -8 + 1 = -7$$ 5. **Conclusion:** The two lines intersect at the point $$(-4, -7)$$.