Graph The Equation And Identify The \[$x\$\]- And \[$y\$\]-intercepts For The Equation: $\[ 2y = -3x - 6 \\]Part 1 Of 5Solve The Equation For \[$y\$\]:$\[ Y = -\frac{3}{2}x - 3 \\]Part 2 Of 5Substitute

Feb 26, 2025 by ADMIN 202 views

**Graphing Equations and Identifying Intercepts: A Step-by-Step Guide**

Understanding the Basics of Graphing Equations

Graphing equations is a fundamental concept in mathematics that involves representing a relationship between two variables, typically x and y, on a coordinate plane. In this article, we will focus on graphing the equation 2y = -3x - 6 and identifying its x- and y-intercepts.

What are Intercepts?

Intercepts are points on a graph where the line crosses the x-axis or the y-axis. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.

Graphing the Equation

To graph the equation 2y = -3x - 6, we need to first solve for y. This will give us the equation in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.

Solving for y

The equation 2y = -3x - 6 can be solved for y by dividing both sides by 2.

# Import necessary modules
import sympy as sp

# Define variables
x = sp.symbols('x')
y = sp.symbols('y')

# Define the equation
equation = 2*y + 3*x + 6

# Solve for y
solution = sp.solve(equation, y)

# Print the solution
print(solution)

This will give us the equation y = -3/2x - 3.

Graphing the Equation

Now that we have the equation in slope-intercept form, we can graph it on a coordinate plane. The graph will be a straight line with a slope of -3/2 and a y-intercept of -3.

Identifying the x- and y-Intercepts

To identify the x- and y-intercepts, we need to find the points where the line crosses the x-axis and the y-axis.

Finding the x-Intercept

The x-intercept is the point where the line crosses the x-axis. To find this point, we need to set y = 0 and solve for x.

# Set y = 0
y = 0

# Solve for x
x_intercept = sp.solve(equation.subs(y, 0), x)

# Print the x-intercept
print(x_intercept)

This will give us the x-intercept.

Finding the y-Intercept

The y-intercept is the point where the line crosses the y-axis. To find this point, we need to set x = 0 and solve for y.

# Set x = 0
x = 0

# Solve for y
y_intercept = sp.solve(equation.subs(x, 0), y)

# Print the y-intercept
print(y_intercept)

This will give us the y-intercept.

Conclusion

In this article, we graphed the equation 2y = -3x - 6 and identified its x- and y-intercepts. We solved for y to get the equation in slope-intercept form, which is y = -3/2x - 3. We then graphed the equation on a coordinate plane and identified the x- and y-intercepts by setting y = 0 and solving for x, and setting x = 0 and solving for y.

Step 1: Solve for y

To solve for y, we need to divide both sides of the equation 2y = -3x - 6 by 2.

# Solve for y
y = (-3*x - 6)/2

This will give us the equation y = -3/2x - 3.

Step 2: Graph the Equation

Now that we have the equation in slope-intercept form, we can graph it on a coordinate plane. The graph will be a straight line with a slope of -3/2 and a y-intercept of -3.

Step 3: Identify the x- and y-Intercepts

To identify the x- and y-intercepts, we need to find the points where the line crosses the x-axis and the y-axis.

Step 4: Find the x-Intercept

To find the x-intercept, we need to set y = 0 and solve for x.

# Set y = 0
y = 0

# Solve for x
x_intercept = sp.solve(equation.subs(y, 0), x)

# Print the x-intercept
print(x_intercept)

This will give us the x-intercept.

Step 5: Find the y-Intercept

To find the y-intercept, we need to set x = 0 and solve for y.

# Set x = 0
x = 0

# Solve for y
y_intercept = sp.solve(equation.subs(x, 0), y)

# Print the y-intercept
print(y_intercept)

This will give us the y-intercept.

Conclusion

Frequently Asked Questions

Q: What is the difference between the x-intercept and the y-intercept?

A: The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.

Q: How do I find the x-intercept of a line?

A: To find the x-intercept, set y = 0 and solve for x.

Q: How do I find the y-intercept of a line?

A: To find the y-intercept, set x = 0 and solve for y.

Q: What is the equation of a line in slope-intercept form?

A: The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I graph a line on a coordinate plane?

A: To graph a line on a coordinate plane, use the slope-intercept form of the equation and plot the points on the graph.

Q: What is the significance of the x-intercept and the y-intercept in graphing equations?

A: The x-intercept and the y-intercept are important points on the graph of a line, as they represent the points where the line crosses the x-axis and the y-axis, respectively.

Q: Can I use the x-intercept and the y-intercept to determine the equation of a line?

A: Yes, you can use the x-intercept and the y-intercept to determine the equation of a line. By substituting the x-intercept and the y-intercept into the equation, you can solve for the slope and the y-intercept.

Q: How do I determine the slope of a line using the x-intercept and the y-intercept?

A: To determine the slope of a line using the x-intercept and the y-intercept, use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) is the x-intercept and (x2, y2) is the y-intercept.

Q: Can I use the x-intercept and the y-intercept to graph a line on a coordinate plane?

A: Yes, you can use the x-intercept and the y-intercept to graph a line on a coordinate plane. By plotting the x-intercept and the y-intercept on the graph, you can draw the line.

Common Mistakes to Avoid

Mistake 1: Not setting y = 0 to find the x-intercept

A: Make sure to set y = 0 to find the x-intercept.

Mistake 2: Not setting x = 0 to find the y-intercept

A: Make sure to set x = 0 to find the y-intercept.

Mistake 3: Not using the slope-intercept form of the equation to graph the line

A: Make sure to use the slope-intercept form of the equation to graph the line.

Mistake 4: Not plotting the x-intercept and the y-intercept on the graph

A: Make sure to plot the x-intercept and the y-intercept on the graph.

Conclusion

In this article, we answered frequently asked questions about graphing equations and identifying intercepts. We covered topics such as finding the x-intercept and the y-intercept, graphing a line on a coordinate plane, and determining the slope of a line using the x-intercept and the y-intercept. We also discussed common mistakes to avoid when graphing equations and identifying intercepts.

Step 1: Find the x-intercept

To find the x-intercept, set y = 0 and solve for x.

Step 2: Find the y-intercept

To find the y-intercept, set x = 0 and solve for y.

Step 3: Graph the line on a coordinate plane

To graph the line on a coordinate plane, use the slope-intercept form of the equation and plot the points on the graph.

Step 4: Determine the slope of the line

To determine the slope of the line, use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) is the x-intercept and (x2, y2) is the y-intercept.

Step 5: Plot the x-intercept and the y-intercept on the graph

To plot the x-intercept and the y-intercept on the graph, use the coordinates of the intercepts.

Conclusion

In this article, we provided a step-by-step guide to graphing equations and identifying intercepts. We covered topics such as finding the x-intercept and the y-intercept, graphing a line on a coordinate plane, and determining the slope of a line using the x-intercept and the y-intercept. We also discussed common mistakes to avoid when graphing equations and identifying intercepts.