To graph two intersecting lines based on the equations provided:

1. Labeling the axes:
   The horizontal axis will be the x-axis, and the vertical axis will be the y-axis. These will form the grid for plotting the lines.

2. Plotting the first line:
   The first equation is $y = 2x + 3$.

   • The slope is 2, meaning that for every 1 unit increase in $x$, $y$ increases by 2 units.
   • The y-intercept is 3, which means the line crosses the y-axis at $(0, 3)$.

   To plot, start at $(0, 3)$ and use the slope $2$ (rise over run) to plot a few more points. For example, at $x = 1$, $y = 2(1) + 3 = 5$, so the point $(1, 5)$ is on the line.

3. Plotting the second line:
   The second equation is $y = -x + 1$.

   • The slope is -1, meaning that for every 1 unit increase in $x$, $y$ decreases by 1 unit.
   • The y-intercept is 1, so the line crosses the y-axis at $(0, 1)$.

   To plot, start at $(0, 1)$ and use the slope $-1$. For example, at $x = 1$, $y = -(1) + 1 = 0$, so the point $(1, 0)$ is on the line.

4. Finding the intersection point:
   To find where the lines intersect, solve the system of equations:

   • $y = 2x + 3$
   • $y = -x + 1$

   Setting them equal: $2x + 3 = -x + 1$ Solve for $x$: $2x + x = 1 - 3$ $3x = -2 \quad \Rightarrow \quad x = -\frac{2}{3}$ Substitute $x = -\frac{2}{3}$ back into either equation, say $y = 2x + 3$: $y = 2(-\frac{2}{3}) + 3 = -\frac{4}{3} + 3 = \frac{5}{3}$ So, the intersection point is $\left( -\frac{2}{3}, \frac{5}{3} \right)$.

5. Final Graph:
   Now plot both lines, ensuring you label the intersection point $\left( -\frac{2}{3}, \frac{5}{3} \right)$. The axes should be labeled with the appropriate units.

Would you like a more detailed breakdown of the graphing process or any clarification on the algebraic steps? Also, here are a few related questions for further exploration:

1. How would the graph change if the slopes of the lines were both positive?
2. What would happen if the two lines had the same slope? How would this affect the solution?
3. How can you graph a system with three equations?
4. How do the solutions of a system of equations relate to the graphs of the lines?
5. What methods can be used to solve a system of linear equations without graphing?

Tip: When graphing lines, always check the slope and intercept first, and plot key points before connecting them to visualize the relationship.