To graph the equations $x + y = 5$ and $y = 3x - 7$ on the same set of axes, we need to plot each line and find their point of intersection. Here are the steps:

1. Graphing $x + y = 5$:

   • Solve for $y$: $y = 5 - x$
   • Find two points on the line:
     • When $x = 0$, $y = 5$ → Point: $(0, 5)$
     • When $y = 0$, $x = 5$ → Point: $(5, 0)$

2. Graphing $y = 3x - 7$:

   • Find two points on the line:
     • When $x = 0$, $y = -7$ → Point: $(0, -7)$
     • When $y = 0$, solve $0 = 3x - 7$ → $3x = 7$ → $x = \frac{7}{3} \approx 2.33$ → Point: $(\frac{7}{3}, 0)$

3. Find the point of intersection:

   • Set $y$ values equal to each other: $5 - x = 3x - 7$
   • Solve for $x$: $5 - x = 3x - 7$ $5 + 7 = 3x + x$ $12 = 4x$ $x = 3$
   • Substitute $x = 3$ into one of the equations to find $y$: $y = 5 - 3 = 2$

   Therefore, the point of intersection is $(3, 2)$.

Now, let's plot the graphs and mark the point of intersection:

$\text{Graphing the lines and marking the intersection point:}$

x & y = 5 - x & y = 3x - 7 \\ \hline 0 & 5 & -7 \\ 5 & 0 & 8 \\ \end{array}$$ ```python import matplotlib.pyplot as plt import numpy as np # Define the range for x x = np.linspace(-2, 6, 400) # Define the equations y1 = 5 - x y2 = 3*x - 7 # Create the plot plt.figure(figsize=(8, 6)) # Plot the equations plt.plot(x, y1, label='$x + y = 5$') plt.plot(x, y2, label='$y = 3x - 7$') # Plot the point of intersection plt.plot(3, 2, 'ro') # red dot plt.text(3, 2, ' (3, 2)', fontsize=12, verticalalignment='bottom') # Set the labels and title plt.xlabel('x') plt.ylabel('y') plt.title('Graphs of $x + y = 5$ and $y = 3x - 7$') plt.axhline(0, color='black',linewidth=0.5) plt.axvline(0, color='black',linewidth=0.5) plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5) plt.legend() # Show the plot plt.show() ``` The graph shows the lines intersecting at the point $$(3, 2)$$. Would you like more details or have any questions about this? Here are some related questions you might find useful: 1. What are the steps to graph linear equations manually? 2. How do you find the slope of a line given its equation? 3. How can you determine if two lines are parallel or perpendicular? 4. What is the significance of the y-intercept in a linear equation? 5. How do you solve systems of linear equations algebraically? 6. What are some applications of finding the intersection of two lines? 7. How can you check your graphical solution algebraically? 8. What is the effect of changing the slope or y-intercept on the graph of a line? **Tip:** When graphing lines, always check for a few key points (such as intercepts) to ensure your graph is accurate.