Writing Equations for Lines that Pass Through Specific Points

Oct 13, 2025 by thesweetdesigns.com 71 views

Introduction

In mathematics, the slope-intercept form is a fundamental concept used to represent lines on a coordinate plane. This form is expressed as y = mx + b, where m represents the slope of the line and b is the y-intercept. In this article, we will explore how to write equations in slope-intercept form for lines that pass through specific points and are parallel or perpendicular to given lines.

Writing Equations for Lines that Pass Through Specific Points

22. Perpendicular to y = (1/2)x + 9, Passes Through (-7, -4)

To write an equation for a line that is perpendicular to y = (1/2)x + 9 and passes through the point (-7, -4), we need to find the slope of the given line. The slope of y = (1/2)x + 9 is 1/2. Since the line we are looking for is perpendicular, its slope will be the negative reciprocal of 1/2, which is -2.

The equation of the line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. We know the slope (m = -2) and the point (-7, -4), which we can use to find the y-intercept (b).

Using the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point, we can substitute the values:

y - (-4) = -2(x - (-7)) y + 4 = -2(x + 7) y + 4 = -2x - 14 y = -2x - 18

Therefore, the equation of the line in slope-intercept form is y = -2x - 18.

23. Parallel to y = 7, Passes Through (-1, -10)

To write an equation for a line that is parallel to y = 7 and passes through the point (-1, -10), we need to find the slope of the given line. The slope of y = 7 is 0, since it is a horizontal line.

Since the line we are looking for is parallel, it will have the same slope as the given line, which is 0.

The equation of the line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. We know the slope (m = 0) and the point (-1, -10), which we can use to find the y-intercept (b).

Using the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point, we can substitute the values:

y - (-10) = 0(x - (-1)) y + 10 = 0 y = -10

Therefore, the equation of the line in slope-intercept form is y = -10.

24. Passes Through (3, 2), Perpendicular to y = 3x - 2

To write an equation for a line that passes through the point (3, 2) and is perpendicular to y = 3x - 2, we need to find the slope of the given line. The slope of y = 3x - 2 is 3.

Since the line we are looking for is perpendicular, its slope will be the negative reciprocal of 3, which is -1/3.

The equation of the line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. We know the slope (m = -1/3) and the point (3, 2), which we can use to find the y-intercept (b).

Using the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point, we can substitute the values:

y - 2 = (-1/3)(x - 3) y - 2 = (-1/3)x + 1 y = (-1/3)x + 3

Therefore, the equation of the line in slope-intercept form is y = (-1/3)x + 3.

Conclusion

In this article, we have explored how to write equations in slope-intercept form for lines that pass through specific points and are parallel or perpendicular to given lines. We have used the point-slope form of a line to find the equation of each line, and have applied the concept of slope and negative reciprocal to determine the slope of the perpendicular line.

Q&A: Frequently Asked Questions about Slope-Intercept Form

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

A: The slope-intercept form of a line is a mathematical expression that represents a line on a coordinate plane. It is expressed as 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 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. The slope is denoted by the letter m.

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. The y-intercept is denoted by the letter b.

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: How do I find the y-intercept of a line?

A: To find the y-intercept of a line, you can use the point-slope form of a line:

y - y1 = m(x - x1)

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

Q: What is the difference between a slope and a negative reciprocal?

A: A slope is a measure of how steep a line is, while a negative reciprocal is the opposite of a slope. If the slope of a line is m, then the negative reciprocal is -1/m.

Q: How do I write an equation for a line that is parallel to a given line?

A: To write an equation for a line that is parallel to a given line, you need to find the slope of the given line and use it as the slope of the new line. The equation of the new line will be in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I write an equation for a line that is perpendicular to a given line?

A: To write an equation for a line that is perpendicular to a given line, you need to find the slope of the given line and use its negative reciprocal as the slope of the new line. The equation of the new line will be in the form y = mx + b, where m is the negative reciprocal of the slope of the given line and b is the y-intercept.

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

A: Yes, you can use the slope-intercept form to find the equation of a line that passes through a specific point. You need to find the slope of the line and use it to write the equation in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: Can I use the slope-intercept form to find the equation of a line that is parallel or perpendicular to a given line?

A: Yes, you can use the slope-intercept form to find the equation of a line that is parallel or perpendicular to a given line. You need to find the slope of the given line and use it to write the equation of the new line in the form y = mx + b, where m is the slope and b is the y-intercept.

Conclusion

In this article, we have answered some of the most frequently asked questions about slope-intercept form. We have covered topics such as the definition of slope-intercept form, how to find the slope and y-intercept of a line, and how to write equations for lines that are parallel or perpendicular to given lines. By following these steps, you can write equations in slope-intercept form for any line that passes through a specific point and is parallel or perpendicular to a given line.