To graph the system of equations and identify its solution, we need to plot the two lines represented by the equations and find the point where they intersect.

System of Equations

1. $2x - 3y = 8$
2. $-3x + y = -5$

Step-by-Step Solution

1. Rearrange each equation to slope-intercept form $y = mx + b$.

Equation 1: $2x - 3y = 8$

Solve for $y$:

$2x - 3y = 8$

Subtract $2x$ from both sides:

$-3y = -2x + 8$

Divide each term by $-3$:

$y = \frac{2}{3}x - \frac{8}{3}$

Equation 2: $-3x + y = -5$

Solve for $y$:

$y = 3x - 5$

2. Graph the equations

Let's plot the lines using these equations:

• The first line is $y = \frac{2}{3}x - \frac{8}{3}$.
• The second line is $y = 3x - 5$.

3. Identify the intersection point

The intersection point of these two lines is the solution to the system of equations. Let's find this point by setting the two equations equal to each other:

$\frac{2}{3}x - \frac{8}{3} = 3x - 5$

Multiply every term by 3 to eliminate fractions:

$2x - 8 = 9x - 15$

Rearrange to find $x$:

$2x - 9x = -15 + 8$

$-7x = -7$

Divide both sides by $-7$:

$x = 1$

Now substitute $x = 1$ back into either equation to solve for $y$. Using the second equation:

$y = 3(1) - 5$

$y = 3 - 5 = -2$

Thus, the solution to the system of equations is $(x, y) = (1, -2)$.

Graph

Here is the graph showing the intersection point:

pythonCopy code
import numpy as np
import matplotlib.pyplot as plt

# Define the range for x
x_values = np.linspace(-5, 5, 400)

# Define the equations
y1_values = (2/3) * x_values - 8/3
y2_values = 3 * x_values - 5

# Create the plot
plt.figure(figsize=(8, 6))
plt.plot(x_values, y1_values, label=r'$2x - 3y = 8$', color='blue')
plt.plot(x_values, y2_values, label=r'$-3x + y = -5$', color='orange')

# Highlight the solution
plt.scatter(1, -2, color='red', zorder=5)
plt.text(1, -2, '(1, -2)', fontsize=12, verticalalignment='bottom')

# Add labels and legend
plt.axhline(0, color='black',linewidth=0.8)
plt.axvline(0, color='black',linewidth=0.8)
plt.title('Graph of the System of Equations')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid(True)
plt.xlim(-5, 5)
plt.ylim(-10, 10)

plt.show()

Explanation

The graph above shows two lines, and their intersection point at $(1, -2)$ is the solution to the system of equations. This is the point where both equations are satisfied simultaneously.

Would you like more details on any of the steps? If you have any questions or need further clarification, feel free to ask!

Further Questions

1. How can we verify the solution algebraically?
2. What happens if the lines are parallel?
3. Can a system of equations have more than one solution?
4. What is the significance of the slope in these equations?
5. How do you determine if a system is consistent or inconsistent?
6. What role does the y-intercept play in graphing equations?
7. How do we handle systems with three or more variables?
8. What are the methods to solve systems of equations other than graphing?

Tip

When graphing lines, always check for parallelism and intersection visually to ensure correct solutions.