Incorrectly identifying the given point and slope

Oct 13, 2025 by windailys.com 24 views

What is the Equation, in Point-Slope Form, for a Line that Goes Through a Given Point and Has a Given Slope?

Understanding the Point-Slope Form of a Line

The point-slope form of a line is a mathematical equation that represents a line in terms of its slope and a point on the line. This form is particularly useful when we know the slope of a line and a point that the line passes through. The point-slope form is given by the equation:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope of the line.

Given Information

We are given a point (8, -4) and a slope of -5/6. We need to find the equation of the line in point-slope form.

Step 1: Identify the Given Point and Slope

The given point is (8, -4) and the slope is -5/6.

Step 2: Plug in the Values into the Point-Slope Form

We will plug in the values of the given point and slope into the point-slope form equation:

y - (-4) = -5/6(x - 8)

Step 3: Simplify the Equation

We will simplify the equation by combining like terms:

y + 4 = -5/6(x - 8)

Step 4: Check the Answer Choices

We will check the answer choices to see if they match the equation we derived:

A. y + 4 = -5/6(x - 8) B. y + 4 = -5/6(x + 8) C. y - 4 = -5/6(x - 8)

Conclusion

The correct answer is A. y + 4 = -5/6(x - 8). This is the equation of the line in point-slope form that passes through the point (8, -4) and has a slope of -5/6.

Why is the Point-Slope Form Important?

The point-slope form is an important concept in mathematics because it allows us to easily find the equation of a line when we know the slope and a point on the line. This form is particularly useful in algebra and geometry, where we often need to find the equation of a line that passes through a given point and has a given slope.

Real-World Applications of the Point-Slope Form

The point-slope form has many real-world applications, including:

Physics: The point-slope form is used to describe the motion of objects under constant acceleration.
Engineering: The point-slope form is used to design and analyze the performance of mechanical systems.
Computer Science: The point-slope form is used in computer graphics to create and manipulate 2D and 3D shapes.

Common Mistakes to Avoid

When working with the point-slope form, there are several common mistakes to avoid:

Incorrectly identifying the given point and slope: Make sure to carefully read and understand the given information.
Incorrectly plugging in values into the point-slope form: Double-check your work to ensure that you have correctly plugged in the values.
Not simplifying the equation: Make sure to simplify the equation to ensure that it is in the correct form.

Conclusion

In conclusion, the point-slope form is an important concept in mathematics that allows us to easily find the equation of a line when we know the slope and a point on the line. By understanding the point-slope form and its applications, we can better solve problems and make informed decisions in a variety of fields.
Frequently Asked Questions (FAQs) about the Point-Slope Form

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

A: The point-slope form of a line is a mathematical equation that represents a line in terms of its slope and a point on the line. It is given by the equation:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope of the line.

Q: How do I use the point-slope form to find the equation of a line?

A: To use the point-slope form, you need to know the slope of the line and a point that the line passes through. You can then plug in the values of the slope and the point into the point-slope form equation and simplify to find the equation of the line.

Q: What is the difference between the point-slope form and the slope-intercept form?

A: The point-slope form and the slope-intercept form are two different ways of representing a line in terms of its slope and a point on the line. The slope-intercept form is given by the equation:

y = mx + b

where m is the slope and b is the y-intercept. The point-slope form is given by the equation:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope of the line.

Q: Can I use the point-slope form to find the equation of a line that is not in the standard form?

A: Yes, you can use the point-slope form to find the equation of a line that is not in the standard form. However, you need to make sure that you have correctly identified the slope and the point on the line.

Q: How do I check if the equation I derived using the point-slope form is correct?

A: To check if the equation you derived using the point-slope form is correct, you can plug in the values of the slope and the point into the equation and simplify. If the equation is correct, it should match the original equation.

Q: Can I use the point-slope form to find the equation of a line that has a negative slope?

A: Yes, you can use the point-slope form to find the equation of a line that has a negative slope. However, you need to make sure that you have correctly identified the slope and the point on the line.

Q: How do I use the point-slope form to find the equation of a line that has a fractional slope?

A: To use the point-slope form to find the equation of a line that has a fractional slope, you need to make sure that you have correctly identified the slope and the point on the line. You can then plug in the values of the slope and the point into the point-slope form equation and simplify to find the equation of the line.

Q: Can I use the point-slope form to find the equation of a line that has a slope of zero?

A: Yes, you can use the point-slope form to find the equation of a line that has a slope of zero. However, you need to make sure that you have correctly identified the slope and the point on the line.

Q: How do I use the point-slope form to find the equation of a line that has a vertical slope?

A: To use the point-slope form to find the equation of a line that has a vertical slope, you need to make sure that you have correctly identified the slope and the point on the line. You can then plug in the values of the slope and the point into the point-slope form equation and simplify to find the equation of the line.

Conclusion

In conclusion, the point-slope form is a powerful tool for finding the equation of a line when you know the slope and a point on the line. By understanding the point-slope form and its applications, you can better solve problems and make informed decisions in a variety of fields.