Write An Equation In Slope-intercept Form For The Line With A { Y $}$-intercept Of 5 And A Slope Of { \frac{2}{3}$} . . . { Y = \frac{2}{3}x + 5 \}

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Introduction

In mathematics, the slope-intercept form of a line is a fundamental concept that helps us understand the relationship between the slope and the y-intercept of a line. The slope-intercept form is represented by the equation y = mx + b, where m is the slope and b is the y-intercept. In this article, we will explore the concept of slope-intercept form and learn how to write an equation for a line with a given y-intercept and slope.

What is Slope-Intercept Form?

The slope-intercept form of a line is a way to represent the equation of a line in a specific format. The equation is written in the form y = mx + b, where:

  • m is the slope of the line
  • b is the y-intercept of the line
  • x is the independent variable
  • y is the dependent variable

The slope-intercept form is useful because it allows us to easily identify the slope and y-intercept of a line from the equation.

Understanding the Components of the Equation

Let's break down the components of the equation y = mx + b:

  • m: The slope of the line. The slope represents the rate of change of the line. A positive slope indicates that the line is increasing, while a negative slope indicates that the line is decreasing.
  • b: The y-intercept of the line. The y-intercept is the point where the line intersects the y-axis. It represents the value of y when x is equal to 0.
  • x: The independent variable. The independent variable is the variable that is not dependent on the other variables in the equation.
  • y: The dependent variable. The dependent variable is the variable that is dependent on the other variables in the equation.

Writing an Equation in Slope-Intercept Form

Now that we understand the components of the equation, let's learn how to write an equation in slope-intercept form. To write an equation in slope-intercept form, we need to know the slope and y-intercept of the line.

Example: Writing an Equation for a Line with a Given Y-Intercept and Slope

Suppose we want to write an equation for a line with a y-intercept of 5 and a slope of 2/3. To do this, we can use the slope-intercept form of the equation, which is y = mx + b.

  • m: The slope of the line is 2/3.
  • b: The y-intercept of the line is 5.

Substituting these values into the equation, we get:

y = (2/3)x + 5

This is the equation of the line in slope-intercept form.

Interpreting the Equation

Now that we have the equation of the line, let's interpret it. The equation y = (2/3)x + 5 tells us that the line has a slope of 2/3 and a y-intercept of 5. This means that for every 1 unit increase in x, the value of y increases by 2/3 units.

Graphing the Line

To graph the line, we can use the equation y = (2/3)x + 5. We can start by plotting the y-intercept, which is the point (0, 5). Then, we can use the slope to find other points on the line.

Conclusion

In this article, we learned how to write an equation in slope-intercept form for a line with a given y-intercept and slope. We also learned how to interpret the equation and graph the line. The slope-intercept form of a line is a powerful tool that helps us understand the relationship between the slope and y-intercept of a line.

Applications of Slope-Intercept Form

The slope-intercept form of a line has many applications in mathematics and real-world problems. Some examples include:

  • Linear equations: The slope-intercept form is used to write linear equations in a specific format.
  • Graphing lines: The slope-intercept form is used to graph lines on a coordinate plane.
  • Solving systems of equations: The slope-intercept form is used to solve systems of linear equations.
  • Real-world problems: The slope-intercept form is used to model real-world problems, such as the cost of goods and services.

Common Mistakes to Avoid

When writing an equation in slope-intercept form, there are several common mistakes to avoid:

  • Incorrect slope: Make sure to use the correct slope value in the equation.
  • Incorrect y-intercept: Make sure to use the correct y-intercept value in the equation.
  • Incorrect equation format: Make sure to use the correct format for the equation, which is y = mx + b.

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the slope-intercept form of a line.

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

A: The slope-intercept form of a line is a way to represent the equation of a line in the format y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between 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 the value of y when x is equal to 0.

Q: How do I write an equation in slope-intercept form?

A: To write an equation in slope-intercept form, you need to know the slope and y-intercept of the line. You can use the equation y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the difference between the slope-intercept form and the standard form of a line?

A: The slope-intercept form (y = mx + b) and the standard form (Ax + By = C) are two different ways to represent the equation of a line. The slope-intercept form is used when the slope and y-intercept are known, while the standard form is used when the equation is in the form of Ax + By = C.

Q: Can I use the slope-intercept form to graph a line?

A: Yes, you can use the slope-intercept form to graph a line. To do this, you need to plot the y-intercept and then use the slope to find other points on the line.

Q: What are some common mistakes to avoid when writing an equation in slope-intercept form?

A: Some common mistakes to avoid when writing an equation in slope-intercept form include:

  • Using the incorrect slope value
  • Using the incorrect y-intercept value
  • Using the incorrect equation format (y = mx + b)

Q: Can I use the slope-intercept form to solve systems of linear equations?

A: Yes, you can use the slope-intercept form to solve systems of linear equations. To do this, you need to write each equation in slope-intercept form and then use the equations to find the solution.

Q: What are some real-world applications of the slope-intercept form?

A: Some real-world applications of the slope-intercept form include:

  • Modeling the cost of goods and services
  • Graphing the relationship between two variables
  • Solving systems of linear equations
  • Finding the equation of a line given two points

Conclusion

In conclusion, the slope-intercept form of a line is a fundamental concept in mathematics that helps us understand the relationship between the slope and y-intercept of a line. By learning how to write an equation in slope-intercept form, we can apply this concept to a variety of real-world problems and mathematical applications.

Additional Resources

For more information on the slope-intercept form, check out the following resources:

  • Khan Academy: Slope-Intercept Form
  • Mathway: Slope-Intercept Form
  • Wolfram Alpha: Slope-Intercept Form

Practice Problems

Try these practice problems to test your understanding of the slope-intercept form:

  1. Write an equation in slope-intercept form for a line with a slope of 2 and a y-intercept of 3.
  2. Graph the line y = 2x + 3.
  3. Solve the system of linear equations: y = 2x + 3 y = x - 2

Answer Key

  1. y = 2x + 3
  2. The graph of the line y = 2x + 3 is a straight line with a slope of 2 and a y-intercept of 3.
  3. The solution to the system of linear equations is x = 1, y = 5.