Select The Correct Answer.A Linear Function Has A $y$-intercept Of -12 And A Slope Of $\frac{3}{2}$. What Is The Equation Of The Line?A. $ Y = \frac{3}{2}x - 12 $B. $ Y = \frac{3}{2}x + 12 $C. $ Y =

Introduction

In mathematics, a linear function is a polynomial function of degree one, which means it has the form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. The equation of a line can be written in two forms: slope-intercept form and standard form. In this article, we will focus on the slope-intercept form, which is $y = mx + b$. We will use this form to find the equation of a line with a given $y$-intercept and slope.

Understanding the Slope-Intercept Form

The slope-intercept form of a linear function is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. The slope, $m$, represents the rate of change of the function, and the $y$-intercept, $b$, represents the point where the line intersects the $y$-axis.

The Equation of a Line with a Given $y$-Intercept and Slope

Given a $y$-intercept of -12 and a slope of $\frac{3}{2}$, we can write the equation of the line in slope-intercept form as $y = \frac{3}{2}x - 12$. This is because the slope-intercept form of a linear function is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Why is the Correct Answer $y = \frac{3}{2}x - 12$?

The correct answer is $y = \frac{3}{2}x - 12$ because it satisfies the given conditions. The slope of the line is $\frac{3}{2}$, which is the coefficient of $x$ in the equation. The $y$-intercept is -12, which is the constant term in the equation.

Why are the Other Options Incorrect?

The other options are incorrect because they do not satisfy the given conditions. Option B, $y = \frac{3}{2}x + 12$, has a positive $y$-intercept, which is not consistent with the given condition. Option C, $y = \frac{3}{2}x - 12$, has the same slope as the correct answer, but the $y$-intercept is not consistent with the given condition.

Conclusion

In conclusion, the equation of a line with a $y$-intercept of -12 and a slope of $\frac{3}{2}$ is $y = \frac{3}{2}x - 12$. This is because the slope-intercept form of a linear function is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. The correct answer satisfies the given conditions, and the other options are incorrect because they do not satisfy the given conditions.

Example Problems

Here are some example problems to help you practice finding the equation of a line with a given $y$-intercept and slope:

• Find the equation of a line with a $y$-intercept of 5 and a slope of \frac{23}.
• Find the equation of a line with a $y$-intercept of -3 and a slope of $\frac{4}{5}$.
• Find the equation of a line with a $y$-intercept of 2 and a slope of $\frac{3}{4}$.

Solutions

Here are the solutions to the example problems:

• The equation of a line with a $y$-intercept of 5 and a slope of $\frac{2}{3}$ is $y = \frac{2}{3}x + 5$.
• The equation of a line with a $y$-intercept of -3 and a slope of $\frac{4}{5}$ is $y = \frac{4}{5}x - 3$.
• The equation of a line with a $y$-intercept of 2 and a slope of $\frac{3}{4}$ is $y = \frac{3}{4}x + 2$.

Tips and Tricks

Here are some tips and tricks to help you find the equation of a line with a given $y$-intercept and slope:

• Make sure to use the slope-intercept form of a linear function, which is $y = mx + b$.
• Use the given slope and $y$-intercept to write the equation of the line.
• Check your answer by plugging in the given values to make sure it satisfies the conditions.

Conclusion

Q: What is a linear function?

A: A linear function is a polynomial function of degree one, which means it has the form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

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

A: The slope-intercept form of a linear function is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Q: How do I find the equation of a line with a given $y$-intercept and slope?

A: To find the equation of a line with a given $y$-intercept and slope, use the slope-intercept form of a linear function, which is $y = mx + b$. Plug in the given slope and $y$-intercept to write the equation of 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 represented by the constant term in the equation of the line.

Q: What is the slope of a line?

A: The slope of a line is the rate of change of the function. It is represented by the coefficient of $x$ in the equation of the line.

Q: How do I determine the equation of a line with a given $y$-intercept and slope?

A: To determine the equation of a line with a given $y$-intercept and slope, use the slope-intercept form of a linear function, which is $y = mx + b$. Plug in the given slope and $y$-intercept to write the equation of the line.

Q: What are some common mistakes to avoid when finding the equation of a line with a given $y$-intercept and slope?

A: Some common mistakes to avoid when finding the equation of a line with a given $y$-intercept and slope include:

• Using the wrong form of the equation of a line.
• Plugging in the wrong values for the slope and $y$-intercept.
• Not checking the answer by plugging in the given values.

Q: How do I check my answer when finding the equation of a line with a given $y$-intercept and slope?

A: To check your answer when finding the equation of a line with a given $y$-intercept and slope, plug in the given values into the equation of the line to make sure it satisfies the conditions.

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

A: Some real-world applications of linear functions include:

• Modeling the cost of producing a product.
• Modeling the revenue of a business.
• Modeling the growth of a population.

Q: How do I use linear functions to model real-world problems?

A: To use linear functions to model real-world problems, identify the variables and the relationships them. Then, use the slope-intercept form of a linear function to write the equation of the line.

Q: What are some tips and tricks for finding the equation of a line with a given $y$-intercept and slope?

A: Some tips and tricks for finding the equation of a line with a given $y$-intercept and slope include:

• Using the slope-intercept form of a linear function.
• Plugging in the given slope and $y$-intercept.
• Checking the answer by plugging in the given values.

Conclusion

In conclusion, finding the equation of a line with a given $y$-intercept and slope is a simple process that involves using the slope-intercept form of a linear function. By following the steps outlined in this article, you can find the equation of a line with a given $y$-intercept and slope. Remember to use the slope-intercept form, use the given slope and $y$-intercept, and check your answer by plugging in the given values.