Why This Equation Represents the Line

Oct 13, 2025 by miodeya.com 133 views

Introduction

In mathematics, the equation of a line can be represented in various forms, including the slope-intercept form, point-slope form, and standard form. When given a point and a slope, we can use the point-slope form to find the equation of the line. In this article, we will explore how to determine the equation of a line that passes through a given point and has a specific slope.

Understanding the Point-Slope Form

The point-slope form of a line is given by the equation:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope of the line. This form is useful when we are given a point and a slope, as it allows us to easily find the equation of the line.

Given Information

We are given a point (-9, -3) and a slope of -6. We need to find the equation of the line that passes through this point and has this slope.

Using the Point-Slope Form

To find the equation of the line, we can use the point-slope form. We will substitute the given point (-9, -3) and the slope (-6) into the equation:

y - y1 = m(x - x1)

y - (-3) = -6(x - (-9))

y + 3 = -6(x + 9)

Evaluating the Options

Now that we have found the equation of the line, we can evaluate the options to determine which one represents the line.

Option A: y - 9 = -6(x - 3)
Option B: y + 9 = -6(x + 3)
Option C: y - 3 = -6(x - 9)
Option D: y + 3 = -6(x + 9)

Conclusion

Based on our analysis, we can see that option D represents the line that passes through the point (-9, -3) and has a slope of -6.

Final Answer

The final answer is:

D. y + 3 = -6(x + 9)

Why This Equation Represents the Line

This equation represents the line because it satisfies the given conditions. The point (-9, -3) lies on the line, and the slope of the line is -6. Therefore, this equation is the correct representation of the line.

Key Takeaways

The point-slope form of a line is given by the equation y - y1 = m(x - x1).
To find the equation of a line that passes through a given point and has a specific slope, we can use the point-slope form.
We can evaluate the options to determine which one represents the line.
The final answer is D. y + 3 = -6(x + 9).

Additional Examples

Here are some additional examples of finding the equation of a line that passes through a given point and has a specific slope:

Find the equation of the line that passes through the point (2, 4) and has a slope of 3.
Find the equation of the line that passes through the point (-2, 1) and has a slope of -2.
Find the equation of the line that passes through the point (5, 2) and has a slope of 1.

Solutions to Additional Examples

The equation of the line that passes through the point (2, 4) and has a slope of 3 is y - 4 = 3(x - 2).
The equation of the line that passes through the point (-2, 1) and has a slope of -2 is y - 1 = -2(x + 2).
The equation of the line that passes through the point (5, 2) and has a slope of 1 is y - 2 = 1(x - 5).

Conclusion

Introduction

In our previous article, we explored how to find the equation of a line that passes through a given point and has a specific slope using the point-slope form. In this article, we will answer some frequently asked questions related to this topic.

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

A: The point-slope form of a line is given by the equation:

y - y1 = m(x - x1)

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

Q: How do I find the equation of a line that passes through a given point and has a specific slope?

A: To find the equation of a line that passes through a given point and has a specific slope, you can use the point-slope form. Simply substitute the given point and the slope into the equation:

y - y1 = m(x - x1)

Q: What if the point is not in the standard form (x, y)?

A: If the point is not in the standard form (x, y), you can still use the point-slope form. Simply substitute the given point into the equation:

y - y1 = m(x - x1)

For example, if the point is (-9, -3), you can substitute it into the equation as follows:

y - (-3) = -6(x - (-9))

Q: How do I evaluate the options to determine which one represents the line?

A: To evaluate the options, simply substitute the given point and the slope into each option and see which one satisfies the equation. For example, if the point is (-9, -3) and the slope is -6, you can substitute it into each option as follows:

Option A: y - 9 = -6(x - 3)
Option B: y + 9 = -6(x + 3)
Option C: y - 3 = -6(x - 9)
Option D: y + 3 = -6(x + 9)

Q: What if I get a different equation than the one I expected?

A: If you get a different equation than the one you expected, it's possible that you made a mistake in substituting the point and the slope into the equation. Double-check your work and make sure that you substituted the correct values.

Q: Can I use the point-slope form to find the equation of a line that passes through two points?

A: No, the point-slope form is used to find the equation of a line that passes through a single point and has a specific slope. If you want to find the equation of a line that passes through two points, you can use the two-point form.

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

A: The two-point form of a line is given by the equation:

y - y1 = m(x - x1)

where (x1, y1) and (x2, y2) are two points on the line.

Q: How do I find the equation of a line that passes through two points using the two-point form?

A: To find the equation of a line that passes through two points using the two-point form, you can use the following steps:

Find the slope of the line using the formula:

m = (y2 - y1) / (x2 - x1) 2. Substitute the slope and one of the points into the equation:

y - y1 = m(x - x1) 3. Simplify the equation to find the final equation of the line.

Conclusion

In conclusion, we have answered some frequently asked questions related to finding the equation of a line that passes through a given point and has a specific slope. We have also explored the point-slope form and the two-point form of a line.