Convert The Equation To Slope-intercept Form, Find The \[$y\$\]-intercept, And Identify The Points.Given: $\[ 3y - 2x = 9 \\]1. Slope-intercept Form (\[$y = Mx + B\$\]): - \[$y = \frac{2}{3}x + 3\$\] (Option E)2.
Introduction
In mathematics, the slope-intercept form of a linear equation is a powerful tool for understanding the relationship between the variables x and y. It is represented as y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will explore how to convert a given equation to slope-intercept form, find the y-intercept, and identify the points on the line.
Converting the Equation to Slope-Intercept Form
To convert the given equation 3y - 2x = 9 to slope-intercept form, we need to isolate y on one side of the equation. We can start by adding 2x to both sides of the equation:
3y = 2x + 9
Next, we can divide both sides of the equation by 3 to solve for y:
y = (2x + 9) / 3
Now, we can simplify the equation by distributing the 1/3 to the terms inside the parentheses:
y = (2/3)x + 3
This is the slope-intercept form of the equation, where m = 2/3 and b = 3.
Finding the y-Intercept
The y-intercept is the point where the line intersects the y-axis. In the slope-intercept form of the equation, the y-intercept is represented by the constant term b. In this case, the y-intercept is 3.
Identifying the Points on the Line
To identify the points on the line, we can use the slope-intercept form of the equation. We can start by substituting different values of x into the equation and solving for y. For example, if we substitute x = 0, we get:
y = (2/3)(0) + 3 y = 3
So, the point (0, 3) is on the line. We can also substitute x = 3 to get:
y = (2/3)(3) + 3 y = 5
So, the point (3, 5) is also on the line.
Discussion
Converting an equation to slope-intercept form is an important skill in mathematics. It allows us to understand the relationship between the variables x and y and to identify the points on the line. In this article, we have seen how to convert a given equation to slope-intercept form, find the y-intercept, and identify the points on the line.
Conclusion
In conclusion, converting an equation to slope-intercept form is a powerful tool for understanding the relationship between the variables x and y. It allows us to identify the points on the line and to understand the slope and y-intercept of the line. By following the steps outlined in this article, we can convert any linear equation to slope-intercept form and gain a deeper understanding of the mathematics behind it.
Example Problems
- Convert the equation 2y + 5x = 7 to slope-intercept form.
- Find the y-intercept of the equation y = 2x + 3.
- Identify the points on the line represented by the equation y = x - 2.
Answer Key
- y = (-5/2)x + 7/2
- The y-intercept is 3.
- The points on the line are (0, -2), (1, -1), and (2, 0).
Glossary
- Slope-intercept form: A mathematical representation of a linear equation in the form y = mx + b, where m is the slope and b is the y-intercept.
- Slope: A measure of the steepness of a line, represented by the coefficient of x in the slope-intercept form of the equation.
- Y-intercept: The point where the line intersects the y-axis, represented by the constant term b in the slope-intercept form of the equation.
Frequently Asked Questions: Converting Equations to Slope-Intercept Form ====================================================================
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is a mathematical representation of a line in the form y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I convert a linear equation to slope-intercept form?
A: To convert a linear equation to slope-intercept form, you need to isolate y on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, and then dividing both sides by the coefficient of y.
Q: What is the y-intercept in the slope-intercept form of an equation?
A: The y-intercept is the point where the line intersects the y-axis. In the slope-intercept form of an equation, the y-intercept is represented by the constant term b.
Q: How do I find the y-intercept of a line?
A: To find the y-intercept of a line, you need to substitute x = 0 into the equation and solve for y. This will give you the y-coordinate of the point where the line intersects the y-axis.
Q: What is the slope of a line in the slope-intercept form of an equation?
A: The slope of a line is represented by the coefficient of x in the slope-intercept form of an equation. It is a measure of the steepness of the line.
Q: How do I identify the points on a line?
A: To identify the points on a line, you can substitute different values of x into the equation and solve for y. This will give you the coordinates of the points on the line.
Q: What are some common mistakes to avoid when converting equations to slope-intercept form?
A: Some common mistakes to avoid when converting equations to slope-intercept form include:
- Not isolating y on one side of the equation
- Not dividing both sides of the equation by the coefficient of y
- Not checking for extraneous solutions
- Not using the correct order of operations
Q: How do I check my work when converting equations to slope-intercept form?
A: To check your work when converting equations to slope-intercept form, you can:
- Plug in different values of x into the equation and solve for y
- Graph the equation and check if it matches the slope-intercept form
- Use a calculator to check if the equation is in slope-intercept form
Q: What are some real-world applications of converting equations to slope-intercept form?
A: Some real-world applications of converting equations to slope-intercept form include:
- Modeling population growth
- Analyzing financial data
- Understanding the relationship between variables in a scientific experiment
Q: Can I use technology to help me convert equations to slope-intercept form?
A: Yes, you can use technology to help you convert equations to slope-intercept form. Some examples include:
- Graphing calculators
- Online graphing tools
- Computer algebra systems
Q: How do I choose the best method for converting equations to slope-intercept form?
A: To choose the best method for converting equations to slope-intercept form, you should consider the following factors:
- The complexity of the equation
- The level of precision required
- The tools and resources available to you
Q: What are some common challenges when converting equations to slope-intercept form?
A: Some common challenges when converting equations to slope-intercept form include:
- Difficulty isolating y on one side of the equation
- Difficulty dividing both sides of the equation by the coefficient of y
- Difficulty checking for extraneous solutions
Q: How do I overcome common challenges when converting equations to slope-intercept form?
A: To overcome common challenges when converting equations to slope-intercept form, you can:
- Break down the equation into smaller steps
- Use algebraic manipulations to isolate y
- Check your work carefully to avoid errors
Q: What are some tips for mastering the skill of converting equations to slope-intercept form?
A: Some tips for mastering the skill of converting equations to slope-intercept form include:
- Practicing regularly with different types of equations
- Using a variety of tools and resources
- Seeking help from a teacher or tutor when needed