1. **Stating the problem:** We need to find the point of intersection for a system of equations. The point of intersection is where the graphs of the equations meet, meaning the values of $x$ and $y$ satisfy both equations simultaneously. 2. **Formula and rules:** To find the intersection, solve the system by substitution or elimination. The solution $(x,y)$ satisfies both equations. 3. **Example:** Suppose the system is: $$\begin{cases} y = 2x + 3 \\ y = -x + 1 \end{cases}$$ 4. **Set the equations equal:** Since both equal $y$, set: $$2x + 3 = -x + 1$$ 5. **Solve for $x$:** $$2x + 3 = -x + 1$$ $$2x + x = 1 - 3$$ $$3x = -2$$ $$x = \frac{-2}{3}$$ 6. **Substitute $x$ back to find $y$:** $$y = 2\left(\frac{-2}{3}\right) + 3 = \frac{-4}{3} + 3 = \frac{-4}{3} + \frac{9}{3} = \frac{5}{3}$$ 7. **Point of intersection:** $$\left(\frac{-2}{3}, \frac{5}{3}\right)$$ This point satisfies both equations, so it is the solution. **Summary:** To match each system to its intersection, solve each system similarly and identify the $(x,y)$ that satisfies both equations.