Understanding the Equation of a Straight Line
To derive the equation of a straight line, we need to understand its basic components, including the slope and y-intercept.
1. Slope-Intercept Form
- The most common form of a straight line is the slope-intercept form:
y = mx + b
- Here,
- "m" represents the slope of the line.
- "b" is the y-intercept, the point where the line crosses the y-axis.
2. Finding the Slope
- The slope (m) is calculated using two points on the line, (x1, y1) and (x2, y2):
- m = (y2 - y1) / (x2 - x1)
- This formula gives the steepness and direction of the line.
3. Identifying the Y-Intercept
- To find the y-intercept (b), substitute the slope and one point into the slope-intercept equation:
- If you have a point (x1, y1), substitute y1 for y and x1 for x in y = mx + b, then solve for b.
4. Writing the Equation
- Once you have m and b, plug these values back into the slope-intercept form:
- y = mx + b
- This represents the equation of your straight line.
5. Example
- For points (1, 2) and (3, 4):
- Slope m = (4 - 2) / (3 - 1) = 1
- Substitute one point to find b: 2 = 1(1) + b, thus b = 1.
- Final equation: y = 1x + 1 or y = x + 1.
By following these steps, you can derive the equation of any straight line with just two points.