The point-slope form is a linear equation used to represent a line in the coordinate plane. It is particularly useful when you know the slope of a line and a point that lies on the line.
The Point-Slope Formula
The point-slope form of a linear equation is:
y – y1 = m(x – x1)
where:
- (x1, y1) is a point on the line.
- m is the slope of the line.
How to Use the Point-Slope Form
To use the point-slope form, you need to know the slope of the line and the coordinates of a point on the line. Once you have this information, you can plug the values into the formula to get the equation of the line.
Example:
Find the equation of a line that passes through the point (2, 3) and has a slope of 4.
Solution:
Using the point-slope form, we get:
y – 3 = 4(x – 2)
Simplifying the equation, we get:
y – 3 = 4x – 8 y = 4x – 5
Therefore, the equation of the line passing through (2, 3) with a slope of 4 is y = 4x – 5.
Applications of the Point-Slope Form
The point-slope form has many applications in mathematics and other fields. Some common uses include:
- Finding the equation of a line: Given the slope and a point on the line, you can use the point-slope form to find the equation of the line.
- Graphing lines: The point-slope form can be used to graph lines by plotting the given point and using the slope to find additional points on the line.
- Solving word problems: The point-slope form can be used to solve word problems involving linear relationships.
Conclusion
The point-slope form is a valuable tool for understanding and working with linear equations. By understanding the formula and its applications, you can solve a variety of mathematical problems and gain a deeper understanding of lines in the coordinate plane.