A Line Has A Slope Of -3 And A $y$-intercept Of $\frac{1}{6}$. Write Its Equation In Slope-intercept Form.Write Your Answer Using Integers, Proper Fractions, And Improper Fractions In Simplest Form.

Feb 28, 2025 by ADMIN 203 views

A Line in Slope-Intercept Form: Understanding the Equation

Introduction to Slope-Intercept Form

The slope-intercept form of a line is a fundamental concept in mathematics, particularly in algebra and geometry. It is a way to express the equation of a line in a simple and intuitive manner. The slope-intercept form is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will explore the concept of slope-intercept form and how to write the equation of a line with a given slope and y-intercept.

Understanding the Given Information

We are given that a line has a slope of -3 and a y-intercept of 1/6. The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run). In this case, the slope is -3, which means that for every unit of horizontal change, the line will drop by 3 units vertically. The y-intercept, on the other hand, is the point where the line intersects the y-axis. In this case, the y-intercept is 1/6, which means that the line will intersect the y-axis at the point (0, 1/6).

Writing the Equation in Slope-Intercept Form

Now that we have understood the given information, we can write the equation of the line in slope-intercept form. The slope-intercept form is given by the equation y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is -3 and the y-intercept is 1/6, so we can plug these values into the equation to get:

y = -3x + 1/6

Simplifying the Equation

The equation y = -3x + 1/6 is already in its simplest form, but we can simplify it further by converting the fraction to a decimal. To do this, we can divide the numerator (1) by the denominator (6) to get:

y = -3x + 0.17

Conclusion

In this article, we have explored the concept of slope-intercept form and how to write the equation of a line with a given slope and y-intercept. We have seen that the slope-intercept form is given by the equation y = mx + b, where m is the slope and b is the y-intercept. We have also seen how to simplify the equation by converting fractions to decimals. With this knowledge, we can now write the equation of a line in slope-intercept form and understand the relationship between the slope and y-intercept.

Examples and Applications

Here are a few examples and applications of the concept of slope-intercept form:

Example 1: Find the equation of a line with a slope of 2 and a y-intercept of 3.
Example 2: Find the equation of a line with a slope of -1 and a y-intercept of 2.
Example 3: Find the equation of a line with a slope of 1/2 and a y-intercept of 1.

Solutions to Examples

Example 1: The equation of the line is y = 2x + 3.
Example 2: The equation of the line is y = -x + 2.
Example 3: The equation of the line is y = 1/2x + 1.

Real-World Applications

The concept of slope-intercept form has many real-world applications, including:

Physics: The slope-intercept form is used to describe the motion of objects under the influence of gravity.
Engineering: The slope-intercept form is used to design and build roads, bridges, and other infrastructure.
Economics: The slope-intercept form is used to model the relationship between economic variables, such as supply and demand.

Conclusion

In conclusion, the concept of slope-intercept form is a fundamental idea in mathematics that has many real-world applications. We have seen how to write the equation of a line in slope-intercept form and how to simplify the equation by converting fractions to decimals. With this knowledge, we can now write the equation of a line in slope-intercept form and understand the relationship between the slope and y-intercept.

Introduction

In our previous article, we explored the concept of slope-intercept form and how to write the equation of a line with a given slope and y-intercept. In this article, we will answer some frequently asked questions about slope-intercept form and provide additional examples and explanations.

Q&A

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

A: The slope-intercept form of a line is a way to express the equation of a line in a simple and intuitive manner. It is given by the equation y = mx + b, where m is the slope of the line 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 it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run).

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 find the equation of a line in slope-intercept form?

A: To find the equation of a line in slope-intercept form, you need to know the slope and y-intercept of the line. You can then plug these values into the equation y = mx + b to get the equation of the line.

Q: Can I have a negative slope in slope-intercept form?

A: Yes, you can have a negative slope in slope-intercept form. For example, if the slope is -3, the equation of the line would be y = -3x + b.

Q: Can I have a fraction as a slope in slope-intercept form?

A: Yes, you can have a fraction as a slope in slope-intercept form. For example, if the slope is 1/2, the equation of the line would be y = 1/2x + b.

Q: Can I have a decimal as a slope in slope-intercept form?

A: Yes, you can have a decimal as a slope in slope-intercept form. For example, if the slope is 0.5, the equation of the line would be y = 0.5x + b.

Q: How do I simplify the equation of a line in slope-intercept form?

A: To simplify the equation of a line in slope-intercept form, you can convert fractions to decimals or simplify the equation by combining like terms.

Q: Can I have a negative y-intercept in slope-intercept form?

A: Yes, you can have a negative y-intercept in slope-intercept form. For example, if the y-intercept is -3, the equation of the line would be y = mx - 3.

Q: Can I have a fraction as a y-intercept in slope-intercept form?

A: Yes, you can have a fraction as a y-intercept in slope-intercept form. For example, if the y-intercept is 1/6, the equation of the line would be y = mx + 1/6.

Q: Can I have a decimal as a y-intercept in slope-intercept form?

A: Yes, you can have a decimal as a y-intercept in slope-intercept form. For example, if the y-intercept is 0.5, the equation of the line would be y = mx + 0.5.