Finding the Equation of a Line with a Known Slope and a Given Point

Oct 13, 2025 by pinkandtrash.com 89 views

Introduction

In mathematics, the equation of a line can be determined using the slope-intercept form, which is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. When we are given a point that lies on the line and the slope of the line, we can use this information to find the equation of the line. In this article, we will explore how to find the equation of a line that passes through a given point and has a known slope.

Understanding the Slope-Intercept Form

The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line is a measure of how steep the line is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The y-intercept is the point at which the line intersects the y-axis.

Finding the Equation of a Line with a Known Slope and a Given Point

To find the equation of a line that passes through a given point and has a known slope, we can use the point-slope form of a line, which 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. We can then substitute the values of the given point and the slope into this equation to find the equation of the line.

Example Problem

Let's consider an example problem to illustrate how to find the equation of a line that passes through a given point and has a known slope. Suppose we are given the point (-1, -5) and the slope of the line is -2. We can use the point-slope form of a line to find the equation of the line.

Step 1: Write Down the Point-Slope Form of a Line

The point-slope form of a line 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. In this case, the given point is (-1, -5) and the slope of the line is -2.

Step 2: Substitute the Values of the Given Point and the Slope into the Equation

We can substitute the values of the given point and the slope into the point-slope form of a line to get the equation y - (-5) = -2(x - (-1)).

Step 3: Simplify the Equation

We can simplify the equation by combining like terms. The equation y - (-5) = -2(x - (-1)) can be rewritten as y + 5 = -2(x + 1).

Step 4: Rewrite the Equation in Slope-Intercept Form

We can rewrite the equation in slope-intercept form by isolating the variable y. The equation y + 5 = -2(x + 1) can be rewritten as y = -2x - 7.

Conclusion

In this article, we have explored how to find the equation of a line that passes through a given point and has a known slope. We have used the point-slope form of a line to find the equation of the line, and we have simplified the equation to get the equation in slope-intercept form. The equation of the line that passes through the point (-1, -5) and has a slope of -2 is y = -2x - 7.

Discussion

The equation of a line can be determined using the slope-intercept form, which is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. When we are given a point that lies on the line and the slope of the line, we can use this information to find the equation of the line. In this article, we have used the point-slope form of a line to find the equation of the line, and we have simplified the equation to get the equation in slope-intercept form.

Final Answer

The final answer is y = -2x - 7.

Introduction

In our previous article, we explored how to find the equation of a line that passes through a given point and has a known slope. In this article, we will answer some frequently asked questions about finding the equation of a line.

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 the given point 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 known slope?

A: To find the equation of a line that passes through a given point and has a known slope, you can use the point-slope form of a line. Substitute the values of the given point and the slope into the equation, and then simplify the equation to get the equation in slope-intercept form.

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

A: The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.

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

A: To find the y-intercept of a line, you can use the slope-intercept form of a line. The y-intercept is the value of b in the equation y = mx + b.

Q: What is the difference between the point-slope form and the slope-intercept 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 the given point and m is the slope of the line. The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. The point-slope form is used to find the equation of a line that passes through a given point and has a known slope, while the slope-intercept form is used to find the equation of a line that has a known slope and y-intercept.

Q: Can I use the point-slope form to find the equation of a line that has a known slope and y-intercept?

A: No, you cannot use the point-slope form to find the equation of a line that has a known slope and y-intercept. The point-slope form is used to find the equation of a line that passes through a given point and has a known slope, while the slope-intercept form is used to find the equation of a line that has a known slope and y-intercept.

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

A: To determine 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 significance of the slope of a line?

A: The slope of a line is a measure of how steep the line is. A line with a positive slope rises from left to right, while a line with a negative slope falls from left to right.

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

A: Yes, you can use the slope-intercept form to find the equation of a line that passes through a given point and has a known slope. However, you will need to use the point-slope form to find the equation of the line first, and then simplify the equation to get the equation in slope-intercept form.

Conclusion

In this article, we have answered some frequently asked questions about finding the equation of a line. We have discussed the point-slope form and the slope-intercept form of a line, and we have provided examples of how to use these forms to find the equation of a line. We hope that this article has been helpful in answering your questions about finding the equation of a line.

Final Answer

The final answer is that the equation of a line can be found using the point-slope form or the slope-intercept form, depending on the information that is given.