Graph The Line With The Equation Y = − 2 3 X − 1 Y = -\frac{2}{3}x - 1 Y = − 3 2 ​ X − 1 .

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Introduction

Graphing linear equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on graphing the line with the equation y=−23x−1y = -\frac{2}{3}x - 1. We will break down the process into simple steps, making it easy to understand and follow.

Understanding the Equation

Before we start graphing, let's understand the equation y=−23x−1y = -\frac{2}{3}x - 1. This is a linear equation in the slope-intercept form, where yy is the dependent variable, and xx is the independent variable. The slope of the line is −23-\frac{2}{3}, and the y-intercept is −1-1.

Graphing the Line

To graph the line, we need to find two points on the line. We can do this by substituting different values of xx into the equation and solving for yy. Let's start by finding the y-intercept, which is the point where the line intersects the y-axis.

Finding the Y-Intercept

The y-intercept is the point where x=0x = 0. Substituting x=0x = 0 into the equation, we get:

y=−23(0)−1y = -\frac{2}{3}(0) - 1 y=−1y = -1

So, the y-intercept is the point (0,−1)(0, -1).

Finding Another Point

Now, let's find another point on the line. We can do this by substituting a value of xx into the equation and solving for yy. Let's choose x=3x = 3.

y=−23(3)−1y = -\frac{2}{3}(3) - 1 y=−2−1y = -2 - 1 y=−3y = -3

So, the point (3,−3)(3, -3) is on the line.

Plotting the Points

Now that we have two points on the line, we can plot them on a coordinate plane. The y-intercept is the point (0,−1)(0, -1), and the other point is (3,−3)(3, -3).

Drawing the Line

With the two points plotted, we can draw the line. The line passes through the two points, and it has a slope of −23-\frac{2}{3}.

Interpreting the Graph

The graph of the line y=−23x−1y = -\frac{2}{3}x - 1 represents a straight line with a negative slope. The line passes through the y-intercept at (−1,0)(-1, 0) and has a slope of −23-\frac{2}{3}. This means that for every unit increase in xx, the value of yy decreases by 23\frac{2}{3} units.

Real-World Applications

Graphing linear equations has many real-world applications. For example, in physics, the equation of motion for an object under constant acceleration is a linear equation. In economics, the demand curve for a product is often represented by a linear equation. In engineering, the equation of a straight line is used to design and build structures such as bridges and buildings.

Conclusion

Graphing linear equations is a fundamental concept in mathematics, and has many real-world applications. In this article, we graphed the line with the equation y=−23x−1y = -\frac{2}{3}x - 1 and interpreted the graph. We also discussed the real-world applications of graphing linear equations. With this knowledge, you can now graph linear equations with ease and apply them to various fields.

Additional Resources

For more information on graphing linear equations, check out the following resources:

  • Khan Academy: Graphing Linear Equations
  • Mathway: Graphing Linear Equations
  • Wolfram Alpha: Graphing Linear Equations

Frequently Asked Questions

Q: What is the slope of the line y=−23x−1y = -\frac{2}{3}x - 1? A: The slope of the line is −23-\frac{2}{3}.

Q: What is the y-intercept of the line y=−23x−1y = -\frac{2}{3}x - 1? A: The y-intercept is −1-1.

Q: How do I graph a linear equation? A: To graph a linear equation, find two points on the line and plot them on a coordinate plane. Then, draw the line passing through the two points.

Glossary

  • Slope: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change to the horizontal change.
  • Y-intercept: The y-intercept is the point where the line intersects the y-axis.
  • Linear equation: A linear equation is an equation in which the highest power of the variable is 1.
    Graphing Linear Equations: A Q&A Guide =====================================

Introduction

Graphing linear equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will provide a comprehensive Q&A guide on graphing linear equations. We will cover various topics, including the slope-intercept form, graphing lines, and real-world applications.

Q&A

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

A: The slope-intercept form of a linear equation is a way of writing an equation in which the highest power of the variable is 1. It is written in the form y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change to the horizontal change. The slope is denoted by the letter mm.

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 y-intercept of a line?

A: The y-intercept of a line is the point where the line intersects the y-axis. It is denoted by the letter bb.

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

A: To find the y-intercept of a line, you can set x=0x = 0 in the equation and solve for yy.

Q: How do I graph a linear equation?

A: To graph a linear equation, you can use the slope-intercept form and plot two points on the line. Then, draw the line passing through the two points.

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 is 1. A quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I determine if a line is parallel or perpendicular to another line?

A: To determine if a line is parallel or perpendicular to another line, you can compare their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular.

Q: What are some real-world applications of graphing linear equations?

A: Graphing linear equations has many real-world applications, including physics, engineering, and economics. For example, in physics, the equation of motion for an object under constant acceleration is a linear equation. In economics, the demand curve for a product is often represented by a linear equation.

Real-World Applications

Graphing linear equations has many real-world applications. Here are a few examples:

  • Physics: The equation of motion for an object under constant acceleration is a linear equation. For example, the equation s=ut+12at2s = ut + \frac{1}{2}at^2 represents the position of an object as a function of time.
  • Engineering: The equation of a straight line is used to design and build structures such as bridges and buildings.
  • Economics: The demand curve for a product is often represented by a linear equation. For example, the equation p=a−bxp = a - bx represents the price of a product as a function of the quantity demanded.

Conclusion

Graphing linear equations is a fundamental concept in mathematics, and it has many real-world applications. In this article, we provided a comprehensive Q&A guide on graphing linear equations, including the slope-intercept form, graphing lines, and real-world applications. We hope that this guide has been helpful in understanding the concept of graphing linear equations.

Additional Resources

For more information on graphing linear equations, check out the following resources:

  • Khan Academy: Graphing Linear Equations
  • Mathway: Graphing Linear Equations
  • Wolfram Alpha: Graphing Linear Equations

Glossary

  • Slope: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change to the horizontal change.
  • Y-intercept: The y-intercept of a line is the point where the line intersects the y-axis.
  • Linear equation: A linear equation is an equation in which the highest power of the variable is 1.
  • Quadratic equation: A quadratic equation is an equation in which the highest power of the variable is 2.