Using the Point to Write the Equation of the Line

Oct 13, 2025 by miodeya.com 101 views

Which Equation Correctly Uses the Point to Write the Equation of the Line in Point-Slope Form?

Understanding Point-Slope Form

The point-slope form of a linear equation is a way to express the equation of a line using the coordinates of a point on the line and the slope of the line. The general form of a point-slope equation is:

y - y1 = m(x - x1)

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

Using the Point to Write the Equation of the Line

To write the equation of a line in point-slope form, we need to know the coordinates of a point on the line and the slope of the line. In this case, we are given the point (-2, -6) and need to determine which equation correctly uses this point to write the equation of the line in point-slope form.

Analyzing the Options

Let's analyze each of the options to determine which one correctly uses the point (-2, -6) to write the equation of the line in point-slope form.

Option A: y - 6 = \frac{5}{2}(x - 2)

This option has a slope of \frac{5}{2} and a point of (2, 6). However, the point (-2, -6) is not on this line, so this option is not correct.

Option B: y - 6 = \frac{2}{5}(x - 2)

This option has a slope of \frac{2}{5} and a point of (2, 6). However, the point (-2, -6) is not on this line, so this option is not correct.

Option C: y + 6 = \frac{2}{5}(x + 2)

This option has a slope of \frac{2}{5} and a point of (-2, -6). This is the correct option, as the point (-2, -6) is on this line.

Conclusion

The correct equation that uses the point (-2, -6) to write the equation of the line in point-slope form is:

y + 6 = \frac{2}{5}(x + 2)

This equation has a slope of \frac{2}{5} and a point of (-2, -6), making it the correct option.

Understanding the Slope

The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In this case, the slope is \frac{2}{5}, which means that for every 5 units the line travels horizontally, it travels 2 units vertically.

Using the Point-Slope Form to Write the Equation of a Line

The point-slope form of a linear equation is a powerful tool for writing the equation of a line. By using the coordinates of a point on the line and the slope of the line, we can write the equation of the line in a concise and efficient way.

Example

Suppose we want to write the equation of a line that passes through the point (3, 4) and has a slope of 2. We can use the point-slope form to write the equation as:

y - 4 = 2(x - 3)

This equation has a slope of 2 and a point of (3, 4), making it the correct equation for the line.

Conclusion

In conclusion, the point-slope form of a linear equation is a powerful tool for writing the equation of a line. By using the coordinates of a point on the line and the slope of the line, we can write the equation of the line in a concise and efficient way. The correct equation that uses the point (-2, -6) to write the equation of the line in point-slope form is:

y + 6 = \frac{2}{5}(x + 2)

This equation has a slope of \frac{2}{5} and a point of (-2, -6), making it the correct option.

Key Takeaways

The point-slope form of a linear equation is a way to express the equation of a line using the coordinates of a point on the line and the slope of the line.
The general form of a point-slope equation is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
To write the equation of a line in point-slope form, we need to know the coordinates of a point on the line and the slope of the line.
The correct equation that uses the point (-2, -6) to write the equation of the line in point-slope form is y + 6 = \frac{2}{5}(x + 2).
Point-Slope Form Q&A

Frequently Asked Questions About Point-Slope Form

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

A: The point-slope form of a linear equation is a way to express the equation of a line using the coordinates of a point on the line and the slope of the line. The general form of a point-slope equation is 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 write the equation of a line in point-slope form?

A: To write the equation of a line in point-slope form, you need to know the coordinates of a point on the line and the slope of the line. You can use the point-slope form equation y - y1 = m(x - x1) and plug in the values of the point and the slope.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do I find the slope of a line?

A: To find the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Q: What is the point-slope form of a line with a slope of 2 and a point of (3, 4)?

A: The point-slope form of a line with a slope of 2 and a point of (3, 4) is y - 4 = 2(x - 3).

Q: What is the point-slope form of a line with a slope of -3 and a point of (-2, 5)?

A: The point-slope form of a line with a slope of -3 and a point of (-2, 5) is y - 5 = -3(x + 2).

Q: Can I use the point-slope form to write the equation of a vertical line?

A: No, the point-slope form is not suitable for writing the equation of a vertical line. A vertical line has an undefined slope, and the point-slope form requires a defined slope.

Q: Can I use the point-slope form to write the equation of a horizontal line?

A: Yes, you can use the point-slope form to write the equation of a horizontal line. A horizontal line has a slope of 0, and you can plug in the values of the point and the slope into the point-slope form equation.

Q: What are some common mistakes to avoid when using the point-slope form?

A: Some common mistakes to avoid when using the point-slope form include:

Not using the correct coordinates of the point
Not using the correct slope
Not following the correct order of operations
Not simplifying the equation

Q: How do I simplify the equation in point-slope form?

A: To simplify the equation in point-slope form, you can use the following steps:

Combine like terms
Simplify the fraction
Write the equation in the standard form (Ax + By = C)

Q: Can I use the point-slope form to write the equation of a line that passes through the origin?

A: Yes, you can use the point-slope form to write the equation of a line that passes through the origin. The point-slope form equation is y - y1 = m(x - x1), and you can plug in the values of the point (0, 0) and the slope.

Conclusion

The point-slope form of a linear equation is a powerful tool for writing the equation of a line. By using the coordinates of a point on the line and the slope of the line, you can write the equation of the line in a concise and efficient way. Remember to avoid common mistakes, simplify the equation, and use the correct coordinates and slope.