1. **State the problem:** We are given two linear equations: $$y = 3x - 4$$ and $$y = -\frac{1}{2}x + 3$$ We want to find the point where these two lines intersect. 2. **Set the equations equal to find the intersection:** Since both expressions equal $y$, set them equal: $$3x - 4 = -\frac{1}{2}x + 3$$ 3. **Solve for $x$:** Add $\frac{1}{2}x$ to both sides: $$3x + \frac{1}{2}x - 4 = 3$$ Write $3x$ as $\frac{6}{2}x$ to combine: $$\frac{6}{2}x + \frac{1}{2}x - 4 = 3$$ $$\frac{7}{2}x - 4 = 3$$ Add 4 to both sides: $$\frac{7}{2}x = 7$$ Divide both sides by $\frac{7}{2}$: $$x = \frac{7}{\frac{7}{2}} = 7 \times \frac{2}{7} = 2$$ 4. **Find $y$ by substituting $x=2$ into one of the original equations:** Using $y = 3x - 4$: $$y = 3(2) - 4 = 6 - 4 = 2$$ 5. **Conclusion:** The lines intersect at the point $(2, 2)$. This means the solution to the system is: $$\boxed{(2, 2)}$$