Find The Graph Of This Linear Equation: Y = − 1 3 X + 7 Y = -\frac{1}{3}x + 7 Y = − 3 1 X + 7 Click On The Correct Answer:- Graph 1- Graph 2- Graph 3- Graph 4

Mar 9, 2025 by ADMIN 161 views

Linear Equations and Graphs: Understanding the Basics

In mathematics, linear equations are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on finding the graph of a linear equation, specifically the equation $y = -\frac{1}{3}x + 7$ . To do this, we need to understand the concept of slope and y-intercept, which are essential components of a linear equation.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. The slope of a linear equation represents the rate of change of the variable, while the y-intercept represents the point at which the line intersects the y-axis.

Understanding the Slope

The slope of a linear equation is a measure of how steep the line is. It can be positive, negative, or zero. A positive slope indicates that the line slopes upward from left to right, while a negative slope indicates that the line slopes downward from left to right. A slope of zero indicates that the line is horizontal.

Understanding the Y-Intercept

The y-intercept of a linear equation is the point at which the line intersects the y-axis. It is the value of $y$ when $x$ is equal to zero. The y-intercept is an essential component of a linear equation, as it helps to determine the position of the line on the coordinate plane.

Finding the Graph of a Linear Equation

To find the graph of a linear equation, we need to plot the points on the coordinate plane that satisfy the equation. We can do this by substituting different values of $x$ into the equation and solving for $y$ . The resulting points will form a straight line on the coordinate plane.

Graphing the Equation $y = -\frac{1}{3}x + 7$

To graph the equation $y = -\frac{1}{3}x + 7$ , we need to find the points on the coordinate plane that satisfy the equation. We can do this by substituting different values of $x$ into the equation and solving for $y$ .

Let's start by finding the y-intercept. To do this, we set $x$ equal to zero and solve for $y$ .

y = -\frac{1}{3}(0) + 7

y = 7

So, the y-intercept is 7. This means that the line intersects the y-axis at the point (0, 7).

Next, we need to find the slope of the line. The slope is given by the coefficient of $x$ , which is $-\frac{1}{3}$ . This means that the line slopes downward from left to right.

Now, let's find the x-intercept. To do this, we set $y$ equal to zero and solve for $x$ .

0 = -\frac{1}{3}x + 7

\frac{1}{3}x = 7

x = 21

So, the x-intercept is 21. This means that the line intersects the x-axis at the point (21, 0).

Plotting the Points

Now that we have found the y-intercept and x-intercept, we can plot the points on the coordinate plane. We start by plotting the y-intercept at the point (0, 7). Then, we plot the x-intercept at the point (21, 0).

Next, we need to find the slope of the line. We can do this by drawing a line through the two points and measuring the slope. The slope is given by the ratio of the vertical change to the horizontal change.

Let's say we draw a line through the points (0, 7) and (21, 0). The vertical change is 7 units, and the horizontal change is 21 units. The slope is given by the ratio of the vertical change to the horizontal change, which is:

\text{slope} = \frac{\text{vertical change}}{\text{horizontal change}} = \frac{7}{21} = -\frac{1}{3}

So, the slope of the line is $-\frac{1}{3}$ .

Plotting the Line

Now that we have found the slope and plotted the points, we can plot the line on the coordinate plane. We start by drawing a line through the two points (0, 7) and (21, 0). The line should slope downward from left to right, with a slope of $-\frac{1}{3}$ .

Conclusion

In this article, we have learned how to find the graph of a linear equation. We started by understanding the concept of slope and y-intercept, which are essential components of a linear equation. We then graphed the equation $y = -\frac{1}{3}x + 7$ by finding the y-intercept and x-intercept, and plotting the points on the coordinate plane. Finally, we plotted the line on the coordinate plane, using the slope and the points to determine the position of the line.

Graph Options

Now, let's look at the graph options provided:

Graph 1: This graph shows a line with a slope of $-\frac{1}{3}$ and a y-intercept of 7. The line intersects the x-axis at the point (21, 0).
Graph 2: This graph shows a line with a slope of $-\frac{1}{3}$ and a y-intercept of 7. The line intersects the x-axis at the point (21, 0), but the line is not plotted correctly.
Graph 3: This graph shows a line with a slope of $-\frac{1}{3}$ and a y-intercept of 7. The line intersects the x-axis at the point (21, 0), but the line is not plotted correctly.
Graph 4: This graph shows a line with a slope of $-\frac{1}{3}$ and a y-intercept of 7. The line intersects the x-axis at the point (21, 0), but the line is not plotted correctly.

Correct Answer

The correct answer is Graph 1. This graph shows a line with a slope of $-\frac{1}{3}$ and a y-intercept of 7. The line intersects the x-axis at the point (21, 0).

Discussion

This problem requires the student to understand the concept of slope and y-intercept, and to apply this understanding to graph a linear equation. The student must also be able to plot the points on the coordinate plane and determine the position of the line.

Key Concepts

Linear equations
Slope
Y-intercept
Graphing linear equations
Plotting points on the coordinate plane

Mathematical Operations

Substitution
Solving for y
Finding the slope
Plotting the line

Problem-Solving Strategies

Understanding the concept of slope and y-intercept
Applying this understanding to graph a linear equation
Plotting the points on the coordinate plane
Determining the position of the line
Q&A: Linear Equations and Graphs

In the previous article, we discussed how to find the graph of a linear equation. We learned about the concept of slope and y-intercept, and how to apply this understanding to graph a linear equation. In this article, we will answer some frequently asked questions about linear equations and graphs.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It 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 of a linear equation?

A: The slope of a linear equation is a measure of how steep the line is. It can be positive, negative, or zero. A positive slope indicates that the line slopes upward from left to right, while a negative slope indicates that the line slopes downward from left to right. A slope of zero indicates that the line is horizontal.

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

A: The y-intercept of a linear equation is the point at which the line intersects the y-axis. It is the value of $y$ when $x$ is equal to zero.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to find the points on the coordinate plane that satisfy the equation. You can do this by substituting different values of $x$ into the equation and solving for $y$ . The resulting points will form a straight line on the coordinate plane.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, the equation $y = 2x + 3$ is a linear equation, while the equation $y = x^2 + 2x + 1$ is a quadratic equation.

Q: How do I determine the slope of a line?

A: To determine the slope of a line, you need to find the ratio of the vertical change to the horizontal change. You can do this by drawing a line through two points and measuring the vertical and horizontal changes.

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

A: The x-intercept of a linear equation is the point at which the line intersects the x-axis. It is the value of $x$ when $y$ is equal to zero.

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

A: To find the x-intercept of a linear equation, you need to set $y$ equal to zero and solve for $x$ . This will give you the value of $x$ at which the line intersects the x-axis.

Q: What is the difference between a linear equation and a non-linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a non-linear equation is an equation in which the highest power of the variable(s) is greater than 1. For example, the equation $y = 2x + 3$ is a linear equation, while the equation $y = x^2 + 2x + 1$ is a non-linear equation.

Q: How do I graph a non-linear equation?

A: To graph a non-linear equation, you need to find the points on the coordinate plane that satisfy the equation. You can do this by substituting different values of $x$ into the equation and solving for $y$ . The resulting points will form a curve on the coordinate plane.

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

A: The y-intercept of a linear equation is the point at which the line intersects the y-axis. It is the value of $y$ when $x$ is equal to zero. The y-intercept is an essential component of a linear equation, as it helps to determine the position of the line on the coordinate plane.

Q: How do I determine the y-intercept of a linear equation?

A: To determine the y-intercept of a linear equation, you need to set $x$ equal to zero and solve for $y$ . This will give you the value of $y$ at which the line intersects the y-axis.

Q: What is the significance of the slope in a linear equation?

Q: How do I determine the slope of a line?

Conclusion

In this article, we have answered some frequently asked questions about linear equations and graphs. We have discussed the concept of slope and y-intercept, and how to apply this understanding to graph a linear equation. We have also discussed the difference between a linear equation and a non-linear equation, and how to graph a non-linear equation.