On A Piece Of Paper, Graph $y \ \textless \ -\frac{3}{4}x + 2$. Then Determine Which Answer Choice Matches The Graph You Drew.A. Graph A B. Graph B C. Graph C D. Graph D

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.

In our given equation, $y < -\frac{3}{4}x + 2$, we can see that the slope is $-\frac{3}{4}$ and the y-intercept is $2$. The slope tells us how steep the line is, and the y-intercept tells us where the line intersects the y-axis.

Finding the x-Intercept

To find the x-intercept, we need to set $y = 0$ and solve for $x$. Plugging in $y = 0$ into the equation, we get:

$$0 < -\frac{3}{4}x + 2$$

Subtracting $2$ from both sides, we get:

$$-2 < -\frac{3}{4}x$$

Multiplying both sides by $-\frac{4}{3}$, we get:

$$\frac{8}{3} > x$$

So, the x-intercept is $\frac{8}{3}$.

Graphing the Line

Now that we have the slope, y-intercept, and x-intercept, we can graph the line. To do this, we need to plot the y-intercept and then use the slope to find another point on the line.

The y-intercept is $(0, 2)$, so we plot this point on the graph. Then, we use the slope to find another point on the line. The slope is $-\frac{3}{4}$, so we move down $3$ units and to the right $4$ units from the y-intercept. This gives us the point $(\frac{8}{3}, -1)$.

We can continue this process to find more points on the line, but for now, let's just plot the two points we have: $(0, 2)$ and $(\frac{8}{3}, -1)$.

Determining the Graph

Now that we have plotted the two points, we can determine which answer choice matches the graph we drew.

Answer Choices

A. Graph A B. Graph B C. Graph C D. Graph D

Graph A

Graph A shows a line with a slope of $-\frac{3}{4}$ and a y-intercept of $2$. The line passes through the points $(0, 2)$ and $(\frac{8}{3}, -1)$, which matches the graph we drew.

Graph B

Graph B shows a line with a slope of $-\frac{3}{4}$ and a y-intercept of $-2$. The line passes through the points $(0, -2)$ and $(\frac{8}{3}, 1)$, which does not match the graph we drew.

Graph C

Graph C shows a line with a slope of $-\frac{3}{4}$ and a y-intercept of $-1$. The line passes through the points $(0, -1)$ and $(\frac{8}{3}, -2)$, which does not match the graph we drew.

Graph D

Graph D shows a line with a slope of $-\frac{3}{4}$ and a y-intercept of $1$. The line passes through the points $(0, 1)$ and $(\frac{8}{3}, 0)$, which does not match the graph we drew.

Conclusion

Based on our graph, we can see that the correct answer is A. Graph A. This graph shows a line with a slope of $-\frac{3}{4}$ and a y-intercept of $2$, which matches the graph we drew.

Key Takeaways

• To graph a linear equation, we need to find the slope and the y-intercept of the line.
• The slope tells us how steep the line is, and the y-intercept tells us where the line intersects the y-axis.
• To find the x-intercept, we need to set $y = 0$ and solve for $x$.
• To graph the line, we need to plot the y-intercept and then use the slope to find another point on the line.

Practice Problems

1. Graph the equation $y < 2x - 3$.
2. Find the x-intercept of the equation $y = -\frac{2}{3}x + 1$.
3. Graph the equation $y > -\frac{1}{2}x + 2$.

Solutions

1. The graph of the equation $y < 2x - 3$ is a line with a slope of $2$ and a y-intercept of $-3$.
2. The x-intercept of the equation $y = -\frac{2}{3}x + 1$ is $-\frac{3}{2}$.
3. The graph of the equation $y > -\frac{1}{2}x + 2$ is a line with a slope of $-\frac{1}{2}$ and a y-intercept of $2$.
   Graphing Linear Equations: A Q&A Guide =====================================

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 the $x$ term. If the coefficient is positive, the slope is positive. If the coefficient is negative, the slope is negative.

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. The y-intercept is the value of $y$ when $x = 0$.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to find the slope and the y-intercept of the line. Then, you can plot the y-intercept and use the slope to find another point on the line.

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

A: The x-intercept of a linear equation is the value of $x$ when $y = 0$. To find the x-intercept, you need to set $y = 0$ and solve for $x$.

Q: How do I determine which answer choice matches the graph I drew?

A: To determine which answer choice matches the graph you drew, you need to compare the graph you drew with the answer choices. Look for the answer choice that has the same slope and y-intercept as the graph you drew.

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

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

• Not finding the slope and y-intercept of the line
• Not plotting the y-intercept
• Not using the slope to find another point on the line
• Not comparing the graph you drew with the answer choices

Q: How can I practice graphing linear equations?

A: You can practice graphing linear equations by working through practice problems. Try graphing different linear equations and then comparing your graph with the answer choices.

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 between two points
• Modeling the time it takes to complete a task

Q: How can I use graphing linear equations in my everyday life?

A: You can use graphing linear equations in your everyday life by:

• Modeling real-world situations
• Making predictions about the future
• Making decisions based on data
• Creating graphs to visualize data

Q: What are some common types of linear equations?

A: Some common types of linear equations include:

• Slope-intercept form: $y = mx + b$
• Standard form: $Ax + By = C$
• Point-slope form: $y - y_1 = m(x - x_1)$

Q: How can I convert between different forms of linear equations?

A: You can convert between different forms of linear equations by:

• Using algebraic manipulations to rewrite the equation
• Using the slope and y-intercept to find the equation in slope-intercept form
• Using the x and y intercepts to find the equation in standard form

Q: What are some common mistakes to avoid when converting between different forms of linear equations?

A: Some common mistakes to avoid when converting between different forms of linear equations include:

• Not using algebraic manipulations to rewrite the equation
• Not using the slope and y-intercept to find the equation in slope-intercept form
• Not using the x and y intercepts to find the equation in standard form