The point-slope equation is a form of the equation of a line that is particularly useful for writing equations when you know the slope and a point on the line. Here’s a breakdown of the key components:
Key Components:
-
Slope (m): This is the ratio of the rise (vertical change) to the run (horizontal change) between any two points on the line. It indicates how steep the line is.
-
Point (x1, y1): This is a specific point through which the line passes. It serves as a reference point to determine the equation of the line.
Point-Slope Equation:
The point-slope form of the equation of a line is expressed as:
\[ y - y_1 = m(x - x_1) \]
Where:
- \( y \) and \( x \) are the variables representing points along the line.
- \( y_1 \) and \( x_1 \) are the coordinates of a specific point on the line.
- \( m \) is the slope of the line.
Example:
If a line has a slope of 2 and passes through the point (3, 5), the point-slope equation would be:
\[ y - 5 = 2(x - 3) \]
This equation can be rearranged into slope-intercept form \( y = mx + b \) if needed, but the point-slope form is often more convenient when starting with a point and slope.
Slope-Intercept Equation:
Another standard form of a line's equation is the slope-intercept form, represented as:
\[ y = mx + b \]
Where \( b \) is the y-intercept, or the point where the line crosses the y-axis. This form is helpful when you want to easily identify the slope and the y-intercept of the line.