Which Is The Graph Of The Equation $y - 1 = \frac{2}{3}(x - 3$\]?

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 explore the different techniques and methods used to visualize this equation.

Understanding the Equation

The given equation is in the form of a linear equation, which is a polynomial equation of degree one. The general form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this case, the equation is $y - 1 = \frac{2}{3}(x - 3)$, which can be rewritten as $y = \frac{2}{3}x - 2 + 1$.

Slope and Y-Intercept

The slope of the equation is $\frac{2}{3}$, which is a positive value. This means that the graph of the equation will have a positive slope, indicating that it will rise from left to right. The y-intercept of the equation is $-1$, which is the point where the graph intersects the y-axis.

Graphing the Equation

To graph the equation, we can use the slope-intercept form of a linear equation, which is $y = mx + b$. We can start by plotting the y-intercept, which is the point $(0, -1)$. Then, we can use the slope to find another point on the graph. Since the slope is $\frac{2}{3}$, we can move two units to the right and three units up from the y-intercept to find another point on the graph.

Finding the X-Intercept

The x-intercept of the equation is the point where the graph intersects the x-axis. To find the x-intercept, we can set $y = 0$ and solve for $x$. Substituting $y = 0$ into the equation, we get $0 = \frac{2}{3}x - 2 + 1$, which simplifies to $0 = \frac{2}{3}x - 1$. Solving for $x$, we get $x = \frac{3}{2}$.

Graphing the X-Intercept

To graph the x-intercept, we can plot the point $(\frac{3}{2}, 0)$. This point represents the point where the graph intersects the x-axis.

Graphing the Equation

Now that we have found the y-intercept and the x-intercept, we can graph the equation. We can start by plotting the y-intercept, which is the point $(0, -1)$. Then, we can use the slope to find another point on the graph. Since the slope is $\frac{2}{3}$, we can move two units to the right and three units up from the y-intercept to find another point on the graph. We can continue this process to find more points on the graph.

Graphing the Equation Using a Graphing Calculator

If we want to graph the equation using a graphing calculator, we can enter the equation into the calculator and use the graphing function to visualize the equation. The graphing calculator will plot the graph of the equation, and we can use the graph to identify the y-intercept, the x-intercept, and other important features of the graph.

Conclusion

In this article, we have graphed the equation $y - 1 = \frac{2}{3}(x - 3)$ using various techniques and methods. We have found the y-intercept, the x-intercept, and other important features of the graph. We have also used a graphing calculator to visualize the equation. Graphing linear equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics.

Key Takeaways

• The equation $y - 1 = \frac{2}{3}(x - 3)$ can be rewritten as $y = \frac{2}{3}x - 2 + 1$.
• The slope of the equation is $\frac{2}{3}$, which is a positive value.
• The y-intercept of the equation is $-1$, which is the point where the graph intersects the y-axis.
• The x-intercept of the equation is $\frac{3}{2}$, which is the point where the graph intersects the x-axis.
• The graph of the equation can be visualized using a graphing calculator.

References

• [1] "Graphing Linear Equations" by Math Open Reference
• [2] "Graphing Linear Equations" by Khan Academy
• [3] "Graphing Linear Equations" by Wolfram Alpha

Further Reading

• "Graphing Quadratic Equations" by Math Open Reference
• "Graphing Polynomial Equations" by Khan Academy
• "Graphing Rational Equations" by Wolfram Alpha
  Graphing Linear Equations: A Q&A 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 provide a Q&A guide to help you understand and graph linear equations.

Q: What is a linear equation?

A: A linear equation is a polynomial equation of degree one, which can be written in the form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: What is the slope of a linear equation?

A: The slope of a linear equation is the ratio of the change in the y-coordinate to the change in the x-coordinate. It can be calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.

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

A: The y-intercept of a linear equation is the point where the graph intersects the y-axis. It can be calculated using the formula $b = y_1$.

Q: How do I graph a linear equation?

A: To graph a linear equation, you can use the slope-intercept form of a linear equation, which is $y = mx + b$. You can start by plotting the y-intercept, which is the point $(0, b)$. Then, you can use the slope to find another point on the graph.

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

A: To find the x-intercept of a linear equation, you can set $y = 0$ and solve for $x$. This will give you the point where the graph intersects the x-axis.

Q: How do I graph a linear equation using a graphing calculator?

A: To graph a linear equation using a graphing calculator, you can enter the equation into the calculator and use the graphing function to visualize the equation.

Q: What are some common mistakes to avoid when graphing linear equations?

A: Some common mistakes to avoid when graphing linear equations include:

• Not using the correct slope and y-intercept
• Not plotting the y-intercept correctly
• Not using the correct x-intercept
• Not using a graphing calculator correctly

Q: How do I check my work when graphing linear equations?

A: To check your work when graphing linear equations, you can use the following steps:

• Plot the y-intercept and the x-intercept
• Use the slope to find another point on the graph
• Check that the graph is a straight line
• Check that the graph passes through the y-intercept and the x-intercept

Q: What are some real-world applications of graphing linear equations?

A: Some real-world applications of graphing linear equations include:

• Modeling population growth
• Modeling the cost of goods
• Modeling the distance traveled by an object
• Modeling the height of a projectile

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. By following the steps outlined in this Q&A guide, you can learn how to graph linear equations and apply this knowledge to real-world problems.

Key Takeaways

• A linear equation is a polynomial equation of degree one.
• The slope of a linear equation is the ratio of the change in the y-coordinate to the change in the x-coordinate.
• The y-intercept of a linear equation is the point where the graph intersects the y-axis.
• To graph a linear equation, you can use the slope-intercept form of a linear equation.
• To find the x-intercept of a linear equation, you can set $y = 0$ and solve for $x$.

References

• [1] "Graphing Linear Equations" by Math Open Reference
• [2] "Graphing Linear Equations" by Khan Academy
• [3] "Graphing Linear Equations" by Wolfram Alpha

Further Reading

• "Graphing Quadratic Equations" by Math Open Reference
• "Graphing Polynomial Equations" by Khan Academy
• "Graphing Rational Equations" by Wolfram Alpha