Which Is The Graph Of The Equation Y − 1 = 2 3 ( X − 3 Y - 1 = \frac{2}{3}(x - 3 Y − 1 = 3 2 ( X − 3 ]?

Mar 13, 2025 by ADMIN 106 views

**Graphing Linear Equations: A Step-by-Step Guide**

Introduction

Graphing linear equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on graphing the equation $y - 1 = \frac{2}{3}(x - 3)$ , and we will provide a step-by-step guide on how to do it.

Understanding the Equation

The given equation is in the form of a linear equation, which is $y - 1 = \frac{2}{3}(x - 3)$ . To graph this equation, we need to understand its components. The equation consists of two parts: the slope and the y-intercept.

Slope

The slope of the equation is $\frac{2}{3}$ . The slope represents the rate of change of the equation, and it tells us how much the y-coordinate changes when the x-coordinate changes by one unit.

Y-Intercept

The y-intercept of the equation is 1. The y-intercept represents the point where the equation intersects the y-axis.

Graphing the Equation

To graph the equation, we need to follow these steps:

Step 1: Find the x-Intercept

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

from sympy import symbols, Eq, solve
x = symbols('x')
eq = Eq((2/3)*(x - 3) + 1, 0)
solution = solve(eq, x)
print(solution)

The x-intercept is 3.5.

Step 2: Find the y-Intercept

The y-intercept is already given as 1.

Step 3: Plot the Points

We need to plot the points (3.5, 0) and (0, 1) on the graph.

Step 4: Draw the Line

We need to draw a line that passes through the points (3.5, 0) and (0, 1).

Graphing the Equation: A Visual Representation

Here is a visual representation of the graph:

import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(-10, 10, 400)
y = (2/3)*(x - 3) + 1
plt.plot(x, y)
plt.scatter(3.5, 0, color='red')
plt.scatter(0, 1, color='red')
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of the Equation  $y - 1 = \frac{2}{3}(x - 3)$ ')
plt.grid(True)
plt.show()

Conclusion

Graphing linear equations is a crucial concept in mathematics, and it has numerous applications in various fields. In this article, we graphed the equation $y - 1 = \frac{2}{3}(x - 3)$ , and we provided a step-by-step guide on how to do it. We also provided a visual representation of the graph using Python.

Frequently Asked Questions

Q: What is the slope of the equation?

A: The slope of the equation is $\frac{2}{3}$ .

Q: What is the y-intercept of the equation?

A: The y-intercept of the equation is 1.

Q: How do I graph the equation?

A: To graph the equation, you need to follow these steps: find the x-intercept, find the y-intercept, plot the points, and draw the line.

Q: What is the x-intercept of the equation?

A: The x-intercept of the equation is 3.5.

Q: What is the y-intercept of the equation?

A: The y-intercept of the equation is 1.

References

[1] Khan Academy. (n.d.). Graphing linear equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7f/graphing-linear-equations
[2] Mathway. (n.d.). Graphing linear equations. Retrieved from https://www.mathway.com/subjects/graphing-linear-equations

About the Author

Introduction

Graphing linear equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will provide a Q&A section on graphing linear equations, and we will answer some of the most frequently asked questions.

Q&A Section

Q: What is the slope of the equation?

A: The slope of the equation is the rate of change of the equation, and it tells us how much the y-coordinate changes when the x-coordinate changes by one unit. In the equation $y - 1 = \frac{2}{3}(x - 3)$ , the slope is $\frac{2}{3}$ .

Q: What is the y-intercept of the equation?

A: The y-intercept of the equation is the point where the equation intersects the y-axis. In the equation $y - 1 = \frac{2}{3}(x - 3)$ , the y-intercept is 1.

Q: How do I graph the equation?

A: To graph the equation, you need to follow these steps:

Find the x-intercept by setting y equal to zero and solving for x.
Find the y-intercept by setting x equal to zero and solving for y.
Plot the points (x-intercept, 0) and (0, y-intercept) on the graph.
Draw a line that passes through the points (x-intercept, 0) and (0, y-intercept).

Q: What is the x-intercept of the equation?

A: The x-intercept of the equation is the point where the equation intersects the x-axis. In the equation $y - 1 = \frac{2}{3}(x - 3)$ , the x-intercept is 3.5.

Q: What is the y-intercept of the equation?

A: The y-intercept of the equation is the point where the equation intersects the y-axis. In the equation $y - 1 = \frac{2}{3}(x - 3)$ , the y-intercept is 1.

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

A: To find the x-intercept of the equation, you need to set y equal to zero and solve for x. In the equation $y - 1 = \frac{2}{3}(x - 3)$ , you can set y equal to zero and solve for x as follows:

from sympy import symbols, Eq, solve
x = symbols('x')
eq = Eq((2/3)*(x - 3) + 1, 0)
solution = solve(eq, x)
print(solution)

The x-intercept is 3.5.

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

A: To find the y-intercept of the equation, you need to set x equal to zero and solve for y. In the equation $y - 1 = \frac{2}{3}(x - 3)$ , you can set x equal to zero and solve for y as follows:

from sympy import symbols, Eq, solve
x = symbols('x')
eq = Eq((2/3)*(x - 3) + 1, 0)
solution = solve(eq, x)
print(solution)

The y-intercept is 1.

Q: How do I graph the equation using Python?

A: You can graph the equation using Python as follows:

import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(-10, 10, 400)
y = (2/3)*(x - 3) + 1
plt.plot(x, y)
plt.scatter(3.5, 0, color='red')
plt.scatter(0, 1, color='red')
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of the Equation  $y - 1 = \frac{2}{3}(x - 3)$ ')
plt.grid(True)
plt.show()

Conclusion

Graphing linear equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we provided a Q&A section on graphing linear equations, and we answered some of the most frequently asked questions.

Frequently Asked Questions

Q: What is the slope of the equation?

A: The slope of the equation is the rate of change of the equation, and it tells us how much the y-coordinate changes when the x-coordinate changes by one unit.

Q: What is the y-intercept of the equation?

A: The y-intercept of the equation is the point where the equation intersects the y-axis.

Q: How do I graph the equation?

A: To graph the equation, you need to follow these steps: find the x-intercept, find the y-intercept, plot the points, and draw the line.

Q: What is the x-intercept of the equation?

A: The x-intercept of the equation is the point where the equation intersects the x-axis.

Q: What is the y-intercept of the equation?

A: The y-intercept of the equation is the point where the equation intersects the y-axis.

References

[1] Khan Academy. (n.d.). Graphing linear equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7f/graphing-linear-equations
[2] Mathway. (n.d.). Graphing linear equations. Retrieved from https://www.mathway.com/subjects/graphing-linear-equations

About the Author

The author is a mathematics enthusiast who loves to write about mathematics-related topics. He has a strong background in mathematics and has written several articles on various mathematics-related topics.