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

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Understanding the Problem

In this problem, we are given a linear equation in the form of $y\ \textless \ -\frac{3}{4} x+2$. 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 two points on the line. We can do this by substituting different values of x into the equation and solving for y.

Finding the y-Intercept

The y-intercept is the point where the line intersects the y-axis. To find the y-intercept, we set x equal to 0 and solve for y.

y \textless −34(0)+2y\ \textless \ -\frac{3}{4} (0)+2

y \textless 2y\ \textless \ 2

So, the y-intercept is 2.

Finding the x-Intercept

The x-intercept is the point where the line intersects the x-axis. To find the x-intercept, we set y equal to 0 and solve for x.

0 \textless −34x+20\ \textless \ -\frac{3}{4} x+2

−34x+2 \textless 0-\frac{3}{4} x+2\ \textless \ 0

−34x \textless −2-\frac{3}{4} x\ \textless \ -2

x \textless 83x\ \textless \ \frac{8}{3}

So, the x-intercept is 83\frac{8}{3}.

Graphing the Line

Now that we have the y-intercept and the x-intercept, we can graph the line. We start by plotting the y-intercept at (0, 2). Then, we plot the x-intercept at (83,0)\left(\frac{8}{3}, 0\right).

To find the slope of the line, we can use the formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

where (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are two points on the line.

Using the y-intercept and the x-intercept, we get:

m=0−283−0m = \frac{0 - 2}{\frac{8}{3} - 0}

m=−283m = \frac{-2}{\frac{8}{3}}

m=−34m = -\frac{3}{4}

So, the slope of the line is −34-\frac{3}{4}.

Drawing the Line

Now that we have the slope and the y-intercept, we can draw the line. We start by drawing a horizontal line through the y-intercept at (0, 2). Then, we draw a line with a slope of −34-\frac{3}{4} through the y-intercept.

Answer Choices

Now that we have graphed the line, we can compare it to the answer choices.

Graph A

Graph A is a line with a slope of 1 and a y-intercept at (0, 2). This is not the same as our graph, so we can eliminate Graph A.

Graph B

Graph B is a line with a slope of −34-\frac{3}{4} and a y-intercept at (0, 2). This is the same as our graph, so we can choose Graph B.

Graph C

Graph C is a line with a slope of 1 and a y-intercept at (0, 1). This is not the same as our graph, so we can eliminate Graph C.

Graph D

Graph D is a line with a slope of −34-\frac{3}{4} and a y-intercept at (0, 1). This is not the same as our graph, so we can eliminate Graph D.

Conclusion

In this problem, we graphed the linear equation $y\ \textless \ -\frac{3}{4} x+2$ and determined which answer choice matches the graph we drew. We found that the y-intercept is 2 and the x-intercept is 83\frac{8}{3}. We also found that the slope of the line is −34-\frac{3}{4}. Finally, we compared our graph to the answer choices and chose Graph B as the correct answer.

Key Takeaways

  • To graph a linear equation, we need to find two points on the line.
  • We can find the y-intercept by setting x equal to 0 and solving for y.
  • We can find the x-intercept by setting y equal to 0 and solving for x.
  • We can find the slope of the line using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
  • We can draw the line by drawing a horizontal line through the y-intercept and then drawing a line with a slope of mm through the y-intercept.

Practice Problems

  1. Graph the linear equation $y\ \textless \ 2x-3$.
  2. Find the y-intercept and the x-intercept of the line $y\ \textless \ 2x-3$.
  3. Find the slope of the line $y\ \textless \ 2x-3$.
  4. Draw the line $y\ \textless \ 2x-3$.
  5. Compare the graph of the line $y\ \textless \ 2x-3$ to the answer choices and choose the correct answer.
    Graphing Linear Equations: A Q&A Guide =====================================

Frequently Asked Questions

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

Q: What is a linear equation?

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

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to find two points on the line. You can do this by substituting different values of x into the equation and solving for y. Then, you can plot the points on a coordinate plane and draw a line through them.

Q: What is the y-intercept?

A: The y-intercept is the point where the line intersects the y-axis. To find the y-intercept, you set x equal to 0 and solve for y.

Q: What is the x-intercept?

A: The x-intercept is the point where the line intersects the x-axis. To find the x-intercept, you set y equal to 0 and solve for x.

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

A: To find the slope of a line, you can use the formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

where (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are two points on the line.

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

A: The equation of a line in slope-intercept form is:

y=mx+by = mx + b

where mm is the slope and bb is the y-intercept.

Q: How do I graph a line in slope-intercept form?

A: To graph a line in slope-intercept form, you can use the slope and the y-intercept to find two points on the line. Then, you can plot the points on a coordinate plane and draw a line through them.

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

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use the x-intercepts and the vertex to find the points on the graph. Then, you can plot the points on a coordinate plane and draw a curve through them.

Common Mistakes

When graphing linear equations, there are several common mistakes to avoid.

  • Not finding the y-intercept: Make sure to find the y-intercept by setting x equal to 0 and solving for y.
  • Not finding the x-intercept: Make sure to find the x-intercept by setting y equal to 0 and solving for x.
  • Not using the correct slope: Make sure to use the correct slope when graphing the line.
  • Not plotting the points correctly: Make sure to plot the points on the coordinate plane correctly.

Tips and Tricks

When graphing linear equations, here are some tips and tricks to keep in mind.

  • Use a ruler: Use a ruler to draw the line through the points.
  • Use a graphing calculator: Use a graphing calculator to graph the line and find the points.
  • Check your work: Check your work by plugging in different values of x and y to make sure the equation is true.
  • Practice, practice, practice: Practice graphing linear equations to get better at it.

Conclusion

Graphing linear equations can be a challenging task, but with practice and patience, you can become proficient in it. Remember to find the y-intercept and the x-intercept, use the correct slope, and plot the points correctly. With these tips and tricks, you can master the art of graphing linear equations.