The point-slope form is a linear equation that represents a line passing through a given point and having a specific slope. It is a convenient way to express the equation of a line when you know the slope and a point on the line.
The Formula
The point-slope form is:
y - y1 = m(x - x1)
where:
- (x1, y1) are the coordinates of a point on the line.
- m is the slope of the line.
How to Use the Formula
To use the point-slope form, simply plug in the given values for the point and slope. For example, if a line passes through the point (2, 5) and has a slope of 3, the equation of the line would be:
y - 5 = 3(x - 2)
Example
Find the equation of a line that passes through the points (1, 4) and (3, 10).
Solution:
-
Find the slope:
- m = (y2 – y1) / (x2 – x1) = (10 – 4) / (3 – 1) = 6 / 2 = 3
-
Choose a point: We can use either (1, 4) or (3, 10). Let’s use (1, 4).
-
Apply the point-slope form:
- y – 4 = 3(x – 1)
Therefore, the equation of the line is y – 4 = 3(x – 1).
Key Points to Remember
- The point-slope form is a versatile tool for representing linear equations.
- It requires the slope and a point on the line to be known.
- The formula can be used to graph lines, find equations of lines, and solve various geometry problems.
By understanding and applying the point-slope formula, you can effectively work with linear equations in various mathematical contexts.