Find The Graph Of This Linear Equation: ${ Y = -\frac{11}{3}x - 3 }$Click On The Correct Answer:A. Graph 1 B. Graph 2 C. Graph 3 D. Graph 4

Mar 9, 2025 by ADMIN 144 views

**Solving Linear Equations: A Step-by-Step Guide to Finding the Graph**

Introduction

Linear equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for various applications in science, engineering, and economics. In this article, we will focus on finding the graph of a linear equation, which is a crucial step in solving linear equations. We will use the equation $y = -\frac{11}{3}x - 3$ as an example and guide you through the process of finding the correct graph.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. The graph of a linear equation is a straight line, and the slope represents the rate of change of the line.

Understanding the Equation

The given equation is $y = -\frac{11}{3}x - 3$ . To find the graph, we need to understand the slope and y-intercept of the line. The slope is the coefficient of $x$ , which is $-\frac{11}{3}$ . The y-intercept is the constant term, which is $-3$ .

Finding the Graph

To find the graph, we can use the slope-intercept form of a linear equation, which is $y = mx + b$ . We can rewrite the given equation as $y = -\frac{11}{3}x - 3$ . The slope is $-\frac{11}{3}$ , and the y-intercept is $-3$ .

Graphing the Equation

To graph the equation, we can use the slope-intercept form. We can start by plotting the y-intercept, which is $-3$ . Then, we can use the slope to find another point on the line. The slope is $-\frac{11}{3}$ , so we can find another point by moving $-\frac{11}{3}$ units to the right and $-3$ units down from the y-intercept.

Analyzing the Graphs

Now that we have found the graph, we need to analyze the options and determine which one is correct. Let's examine each option:

A. Graph 1

Graph 1 has a slope of $-\frac{11}{3}$ and a y-intercept of $-3$ . This graph matches the equation $y = -\frac{11}{3}x - 3$ .

B. Graph 2

Graph 2 has a slope of $-\frac{11}{3}$ , but the y-intercept is $-2$ . This graph does not match the equation $y = -\frac{11}{3}x - 3$ .

C. Graph 3

Graph 3 has a slope of $-\frac{11}{3}$ , but the y-intercept is $-4$ . This graph does not match the equation $y = -\frac{11}{3}x - 3$ .

D. Graph 4

Graph 4 has a slope of $-\frac{11}{3}$ , but the y-intercept is $-1$ . This graph does not match the equation $y = -\frac{11}{3}x - 3$ .

Conclusion

Based on our analysis, the correct graph is A. Graph 1. This graph has a slope of $-\frac{11}{3}$ and a y-intercept of $-3$ , which matches the equation $y = -\frac{11}{3}x - 3$ .

Tips and Tricks

When graphing a linear equation, make sure to use the slope-intercept form.
The slope represents the rate of change of the line.
The y-intercept is the point where the line intersects the y-axis.
Use the slope to find another point on the line.
Analyze the options and determine which one is correct.

Frequently Asked Questions

Q: What is a linear equation? A: A linear equation is an equation in which the highest power of the variable(s) is 1.
Q: What is the slope-intercept form of a linear equation? A: The slope-intercept form is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
Q: How do I find the graph of a linear equation? A: Use the slope-intercept form and plot the y-intercept. Then, use the slope to find another point on the line.

References

[1] Khan Academy. (n.d.). Linear Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4-linear-equations/x2f1f5-linear-equations-2/x2f1f5-linear-equations-2-article
[2] Mathway. (n.d.). Linear Equations. Retrieved from https://www.mathway.com/subjects/linear-equations

Conclusion

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

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

A: The slope-intercept form is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. This form is useful for graphing linear equations, as it allows us to easily identify the slope and y-intercept.

Q: How do I find the graph of a linear equation?

A: To find the graph of a linear equation, follow these steps:

Identify the slope and y-intercept of the equation.
Plot the y-intercept on a coordinate plane.
Use the slope to find another point on the line.
Draw a line through the two points.

Q: What is the slope of a linear equation?

A: The slope of a linear equation is the coefficient of the variable. In the equation $y = mx + b$ , the slope is $m$ . The slope represents the rate of change of the line.

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

A: The y-intercept of a linear equation is the point where the line intersects the y-axis. In the equation $y = mx + b$ , the y-intercept is $b$ .

Q: How do I determine if a line is parallel or perpendicular to another line?

A: To determine if a line is parallel or perpendicular to another line, follow these steps:

Find the slope of both lines.
If the slopes are equal, the lines are parallel.
If the slopes are negative reciprocals of each other, the lines are perpendicular.

Q: What is the equation of a line that passes through two points?

A: To find the equation of a line that passes through two points, follow these steps:

Find the slope of the line using the two points.
Use the slope-intercept form to write the equation of the line.
Substitute the coordinates of one of the points into the equation to find the y-intercept.

Q: How do I graph a linear equation with a negative slope?

A: To graph a linear equation with a negative slope, follow these steps:

Identify the slope and y-intercept of the equation.
Plot the y-intercept on a coordinate plane.
Use the slope to find another point on the line.
Draw a line through the two points, making sure to extend the line in the correct direction.

Q: What is the equation of a line that passes through the origin?

A: The equation of a line that passes through the origin is $y = mx$ , where $m$ is the slope.

Q: How do I find the equation of a line that passes through a given point and has a given slope?

A: To find the equation of a line that passes through a given point and has a given slope, follow these steps:

Use the point-slope form to write the equation of the line.
Substitute the coordinates of the given point into the equation.
Simplify the equation to find the final form.

Q: What is the equation of a line that is parallel to another line?

A: The equation of a line that is parallel to another line is of the form $y = mx + b$ , where $m$ is the slope of the original line and $b$ is a constant.

Q: How do I find the equation of a line that is perpendicular to another line?

A: To find the equation of a line that is perpendicular to another line, follow these steps:

Find the slope of the original line.
Use the negative reciprocal of the slope to find the slope of the perpendicular line.
Use the point-slope form to write the equation of the perpendicular line.

Conclusion

In conclusion, linear equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for various applications in science, engineering, and economics. By following the steps outlined in this article, you should be able to answer any question related to linear equations. Remember to always identify the slope and y-intercept of the equation, and use the slope-intercept form to find the graph of the line.