Determining the Answer Choice

Oct 13, 2025 by talynt.org 123 views

Understanding the Problem

In this problem, we are given a linear equation in the form of $y < -\frac{3}{4}x + 2$ . Our task is to graph this equation on a piece of paper and then determine which answer choice matches the graph we drew.

Graphing Linear Equations

To graph a linear equation, we need to find the slope and the y-intercept of the line. The slope-intercept form of a linear equation is given by:

y = mx + b

where $m$ is the slope and $b$ is the y-intercept.

Finding the Slope

The slope of the line is given by the coefficient of $x$ in the equation. In this case, the coefficient of $x$ is $-\frac{3}{4}$ , so the slope of the line is $-\frac{3}{4}$ .

Finding the Y-Intercept

The y-intercept of the line is given by the constant term in the equation. In this case, the constant term is $2$ , so the y-intercept of the line is $2$ .

Graphing the Line

To graph the line, we need to plot two points on the line. We can use the slope and the y-intercept to find these points.

First, let's find the point where the line intersects the y-axis. Since the y-intercept is $2$ , the point where the line intersects the y-axis is $(0, 2)$ .

Next, let's find the point where the line intersects the x-axis. Since the slope is $-\frac{3}{4}$ , the line will intersect the x-axis at a point where the y-coordinate is $0$ . We can find this point by substituting $y = 0$ into the equation and solving for $x$ .

0 = -\frac{3}{4}x + 2

Solving for $x$ , we get:

x = \frac{8}{3}

So, the point where the line intersects the x-axis is $(\frac{8}{3}, 0)$ .

Plotting the Points

Now that we have the two points, we can plot them on the graph.

The point $(0, 2)$ is plotted on the y-axis.
The point $(\frac{8}{3}, 0)$ is plotted on the x-axis.

Drawing the Line

With the two points plotted, we can draw the line by connecting the points with a straight line.

Determining the Answer Choice

Now that we have graphed the line, we need to determine which answer choice matches the graph we drew.

Answer Choices

We are given four answer choices: A, B, C, and D. We need to examine each answer choice to see which one matches the graph we drew.

Graph A

Graph A shows a line with a slope of $-\frac{3}{4}$ and a y-intercept of $2$ . However, the line in Graph A is not the same as the line we graphed. The line in Graph A has a different shape and orientation.

Graph B

Graph B shows a line with a slope of $-\frac{3}{4}$ and a y-intercept of $2$ . However, the line in Graph B is not the same as the line we graphed. The line in Graph B has a different shape and orientation.

Graph C

Graph C shows a line with a slope of $-\frac{3}{4}$ and a y-intercept of $2$ . This is the same as the line we graphed. Therefore, Graph C is the correct answer choice.

Graph D

Graph D shows a line with a slope of $-\frac{3}{4}$ and a y-intercept of $2$ . However, the line in Graph D is not the same as the line we graphed. The line in Graph D has a different shape and orientation.

Conclusion

In this problem, we graphed the linear equation $y < -\frac{3}{4}x + 2$ and determined which answer choice matches the graph we drew. We found that Graph C is the correct answer choice.

Key Takeaways

To graph a linear equation, we need to find the slope and the y-intercept of the line.
The slope of the line is given by the coefficient of $x$ in the equation.
The y-intercept of the line is given by the constant term in the equation.
We can plot two points on the line using the slope and the y-intercept.
We can draw the line by connecting the points with a straight line.

Final Answer

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about graphing linear equations.

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

A: The slope-intercept form of a linear equation is given by:

y = mx + b

where $m$ is the slope and $b$ is the y-intercept.

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

A: To find the slope of a linear equation, you need to look at the coefficient of $x$ in the equation. The slope is given by the coefficient of $x$ .

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

A: To find the y-intercept of a linear equation, you need to look at the constant term in the equation. The y-intercept is given by the constant term.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to plot two points on the line. You can use the slope and the y-intercept to find these points.

Q: What are the two points I need to plot on the line?

A: The two points you need to plot on the line are:

The point where the line intersects the y-axis. This point is given by the y-intercept.
The point where the line intersects the x-axis. This point is given by the slope and the y-intercept.

Q: How do I plot the points on the line?

A: To plot the points on the line, you need to use a coordinate plane. The x-axis represents the horizontal direction, and the y-axis represents the vertical direction.

Q: How do I draw the line?

A: To draw the line, you need to connect the two points with a straight line.

Q: What if the line is not a straight line?

A: If the line is not a straight line, it may be a non-linear equation. In this case, you need to use a different method to graph 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 correct points on the line.
Not drawing the line correctly.

Q: How can I practice graphing linear equations?

A: You can practice graphing linear equations by using online graphing tools or by working with a tutor. You can also try graphing different linear equations to see how the slope and y-intercept affect the graph.

Conclusion

In this article, we have answered some of the most frequently asked questions about graphing linear equations. We have covered topics such as the slope-intercept form of a linear equation, finding the slope and y-intercept, and plotting points on the line. We have also discussed common mistakes to avoid and how to practice graphing linear equations.

Key Takeaways

The slope-intercept form of a linear equation is given by $y = mx + b$ .
The slope of a linear equation is given by the coefficient of $x$ .
The y-intercept of a linear equation is given by the constant term.
To graph a linear equation, you need to plot two points on the line.
You can practice graphing linear equations by using online graphing tools or by working with a tutor.

Final Answer

The final answer is that graphing linear equations is an important skill to have in mathematics. By understanding the slope-intercept form of a linear equation, finding the slope and y-intercept, and plotting points on the line, you can graph linear equations with ease.