Point Gradient Formula
A straight line in a cartesian plane passes through an infinite number of points. Each of these points has its own x and y- coordinates. The points a line passes through are used to find its slope. Not only that but such points can also be used to write the equation of a line. One such method is discussed below.
Point Gradient formula
Out of a lot of methods to write/ find/ express the equation of a straight line in a cartesian form, the point-slope or point-gradient formula holds a very significant place in coordinate geometry. As its name suggests, this form of the equation consists of one point that the line passes through and its slope.
Formula
The point-gradient formula is given as follows:
y β y1 = m(x β x1)
Where,
- x and y depict general point coordinates.
- x1 and y1 are the numerical coordinates of a point through which the line passes.
- m represents the slope of the given line.
Derivation of point-gradient formula
Slope of a line passing through two points (x, y) and (x1, y1) = m =
Multiplying both sides by (x β x1), we have:
β y β y1 = m(x β x1)
Hence proved.
Sample Problems
Question 1: What is the equation of a line passing through (2, β4) and slope 5?
Solution:
The given point is (2, β4). Thus, x1 = 2, y1 = β4.
Also, m = slope = 5.
We know the point slope equation of a line is given by y β y1 = m(x β x1).
Substituting the above values in the equation, we have:
y β (-4) = 5(x β 2)
β y + 4 = 5x β 10
β y = 5x β 10 β 4
β y = 5x β 14
Question 2: What is the equation of a line passing through (5, 2) and slope 3/4?
Solution:
The given point is (5, 2). Thus, x1 = 5, y1 = 2.
Also, m = slope = 3/4.
We know the point slope equation of a line is given by y β y1 = m(x β x1).
Substituting the above values in the equation, we have:
y β (2) = 3/4(x β 5)
β y β 2 = 3x/4 β 15/4
β y = 3x/4 β 15/4 + 2
β y = 3x/4 β 7/4
Question 3: What is the equation of a horizontal line passing through (3, 3)?
Solution:
The given point is (3, 3). Thus, x1 = 3, y1 = β3.
Since the slope of a horizontal line is zero, m = 0.
We know the point slope equation of a line is given by y β y1 = m(x β x1).
Substituting the above values in the equation, we have:
y β (3) = 0(x β 3)
β y β 3 = 0
β y = 3
Question 4: It is given that a line passes through the points (1, 1) and (-2, 4). Find its equation using the point-slope formula.
Solution:
We know the point slope equation of a line is given by y β y1 = m(x β x1).
In order to use the point- slope form, we need to calculate the slope of the line first.
Slope = m =
β m = -1
Substituting the above values in the equation, we have:
y β (1) = -1(x β 1)
β y β 1 = -x + 1
β y = -x + 2
Question 5: What is the equation of a line passing through (0, 3) and slope 8?
Solution:
The given point is (0, 3). Thus, x1 = 0, y1 = 3.
Also, m = slope = 8.
We know the point slope equation of a line is given by y β y1 = m(x β x1).
Substituting the above values in the equation, we have:
y β 3 = 8(x β 0)
β y β 3 = 8x
β y = 8x + 3