Graph $y = -\frac{4}{3}x + 8$.

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Introduction

Graphing a linear equation is an essential skill in mathematics, particularly in algebra and geometry. It involves representing a linear equation as a visual graph on a coordinate plane. In this article, we will focus on graphing the linear equation $y = -\frac{4}{3}x + 8$. We will break down the process into manageable steps, making it easy to understand and visualize the graph.

Understanding the Equation

Before we start graphing, let's take a closer look at the equation $y = -\frac{4}{3}x + 8$. This is a linear equation in the slope-intercept form, which is given by $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this equation, the slope is $-\frac{4}{3}$ and the y-intercept is $8$.

The Slope

The slope of a linear equation represents the rate of change of the graph. In this case, the slope is $-\frac{4}{3}$, which means that for every unit increase in $x$, the value of $y$ decreases by $\frac{4}{3}$ units. This indicates that the graph will have a downward slope.

The Y-Intercept

The y-intercept is the point where the graph intersects the y-axis. In this equation, the y-intercept is $8$, which means that the graph will pass through the point $(0, 8)$.

Graphing the Equation

To graph the equation $y = -\frac{4}{3}x + 8$, we can use the slope and y-intercept to find two points on the graph. We can then use these points to draw a line that represents the graph.

Finding the First Point

To find the first point, we can substitute $x = 0$ into the equation. This gives us:

y=−43(0)+8y = -\frac{4}{3}(0) + 8

y=8y = 8

So, the first point is $(0, 8)$.

Finding the Second Point

To find the second point, we can substitute $x = 3$ into the equation. This gives us:

y=−43(3)+8y = -\frac{4}{3}(3) + 8

y=−4+8y = -4 + 8

y=4y = 4

So, the second point is $(3, 4)$.

Drawing the Graph

Now that we have two points on the graph, we can draw a line that represents the graph. We can use a ruler or a straightedge to draw a line that passes through the two points.

Interpreting the Graph

The graph of $y = -\frac{4}{3}x + 8$ is a straight line with a downward slope. The graph passes through the point $(0, 8)$ and has a y-intercept of $8$. The graph also passes through the point $(3, 4)$.

Real-World Applications

Graphing linear equations has many real-world applications. For example, in economics, the graph of a linear equation can be used to represent the relationship between two variables, such as the price of a product and the quantity demanded. In physics, the graph of a linear equation can be used to represent the motion of an object, such as the position of a particle over time.

Conclusion

Graphing a linear equation is an essential skill in mathematics, particularly in algebra and geometry. By understanding the slope and y-intercept of a linear equation, we can graph the equation and interpret the resulting graph. The graph of $y = -\frac{4}{3}x + 8$ is a straight line with a downward slope, passing through the points $(0, 8)$ and $(3, 4)$. This graph has many real-world applications, including economics and physics.

Final Thoughts

Graphing linear equations is a fundamental concept in mathematics, and it has many real-world applications. By understanding the slope and y-intercept of a linear equation, we can graph the equation and interpret the resulting graph. Whether you are a student or a professional, graphing linear equations is an essential skill that can help you solve problems and make informed decisions.

References

  • [1] "Graphing Linear Equations" by Math Open Reference
  • [2] "Linear Equations" by Khan Academy
  • [3] "Graphing Linear Equations" by Purplemath

Glossary

  • Slope: The rate of change of a graph.
  • Y-Intercept: The point where the graph intersects the y-axis.
  • Linear Equation: An equation in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
  • Graph: A visual representation of a linear equation on a coordinate plane.

Introduction

Graphing a linear equation is an essential skill in mathematics, particularly in algebra and geometry. In our previous article, we discussed the process of graphing the linear equation $y = -\frac{4}{3}x + 8$. In this article, we will answer some frequently asked questions about graphing linear equations.

Q&A

Q: What is the slope of the graph of $y = -\frac{4}{3}x + 8$?

A: The slope of the graph of $y = -\frac{4}{3}x + 8$ is $-\frac{4}{3}$.

Q: What is the y-intercept of the graph of $y = -\frac{4}{3}x + 8$?

A: The y-intercept of the graph of $y = -\frac{4}{3}x + 8$ is $8$.

Q: How do I find the x-intercept of the graph of $y = -\frac{4}{3}x + 8$?

A: To find the x-intercept of the graph of $y = -\frac{4}{3}x + 8$, we can set $y = 0$ and solve for $x$. This gives us:

0=−43x+80 = -\frac{4}{3}x + 8

43x=8\frac{4}{3}x = 8

x=843x = \frac{8}{\frac{4}{3}}

x=6x = 6

So, the x-intercept is $(6, 0)$.

Q: How do I find the equation of the line that passes through the points $(0, 8)$ and $(3, 4)$?

A: To find the equation of the line that passes through the points $(0, 8)$ and $(3, 4)$, we can use the slope-intercept form of a linear equation, which is given by $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

First, we need 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 $(x_1, y_1)$ and $(x_2, y_2)$ are the two points.

Plugging in the values, we get:

m=4−83−0m = \frac{4 - 8}{3 - 0}

m=−43m = \frac{-4}{3}

Now that we have the slope, we can find the y-intercept by plugging in one of the points into the equation. Let's use the point $(0, 8)$:

8=−43(0)+b8 = -\frac{4}{3}(0) + b

8=b8 = b

So, the y-intercept is $8$.

Now that we have the slope and y-intercept, we can write the equation of the line:

y=−43x+8y = -\frac{4}{3}x + 8

Q: How do I graph the equation $y = -\frac{4}{3}x + 8$?

A: To graph the equation $y = -\frac{4}{3}x + 8$, we can use the slope and y-intercept to find two points on the graph. We can then use these points to draw a line that represents the graph.

Conclusion

Graphing a linear equation is an essential skill in mathematics, particularly in algebra and geometry. By understanding the slope and y-intercept of a linear equation, we can graph the equation and interpret the resulting graph. In this article, we answered some frequently asked questions about graphing linear equations.

Final Thoughts

Graphing linear equations is a fundamental concept in mathematics, and it has many real-world applications. By understanding the slope and y-intercept of a linear equation, we can graph the equation and interpret the resulting graph. Whether you are a student or a professional, graphing linear equations is an essential skill that can help you solve problems and make informed decisions.

References

  • [1] "Graphing Linear Equations" by Math Open Reference
  • [2] "Linear Equations" by Khan Academy
  • [3] "Graphing Linear Equations" by Purplemath

Glossary

  • Slope: The rate of change of a graph.
  • Y-Intercept: The point where the graph intersects the y-axis.
  • Linear Equation: An equation in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
  • Graph: A visual representation of a linear equation on a coordinate plane.