There are three ways to determine the equation of a straight line. Here, we will discuss the point slop form to find out the equation of a straight line. The point-slope form of the equation of a straight line is represented as:
Point- Slope Form Equation: y – y₁ = m (x – x₁)
The above equation is useful when we know :
- One point on the line i.e (x₁, y₁)
- The slope of a line i.e. ‘m’
What Does The Point In The Equation Stands For?
- (x₁, y₁): These are the known points.
- ‘M’ represents the slope of the line
- (x,y): These are any other points on the line.
Point Slope Form Derivation
Slope (m): Change in yChange in x
This gives y – y₁ x – x₁
Let us start with the slope: y – y₁ x – x₁= m
Rearranging by placing the denominator on the right side of the equation, we get the desired point-slope form equation as: y – y₁ = m (x – x₁)
Let us learn to find the equation of a line using the point-slope form with the help of an example:
Example:
Using the point-slope form, find the equation of a straight line that passes through the point (1,4) with the slope -5.
Solution:
Here,
y₁ = 4
x₁ = 1
m = -5
Using the point-slope form method, we get the equation as:
(y – 4) = -5 (x – 1)
y – 4 = -5x + 5
y = -5x + 5 + 4
y = -5x + 9
Therefore the required equation is y = -5x + 9
Slope Intercept Form
The slope-intercept form is a way to write the equation of a line. The equation in a slope-intercept is represented as:
Slope Intercept – Form : y = mx + b
In the above equation:
- m is the slope
- b is y-intercept
Slope: It measures the steepness of the line. The slope of the line can be calculated by measuring the change in y over the change in x.
Y-intercept : It is the point where the line cur across the y-axis. Here the ordered pair is (0,b).
Let us learn to find the equation of a line using the slope-intercept form with the help of an example: You can also solve other interesting examples on this topic by visiting cuemath website.
Example 1: What will be the equation of a line given the slope 5/3 and a y-intercept ( 0,7)?
Solution:
Here, the y-intercept is (0,7) and slope (m) is 5/3
The slope-intercept form equation is given as:
y = mx + b
Placing the values in the slope-intercept form equation, we get
y = 53(x) + 7
53x- y = – 7
Now, we will resolve the above equation by multiplying with 3 on both sides:
3(53x- y) = 3(-7)
5(x – y) = -21
Therefore, the required equation of a line is: 5(x – y) = -21.
