A line’s slope is a measure of its steepness and direction. It is defined as the change in y coordinate to the change in x coordinate of that line. It is denoted by the symbol m. If two points (x1, y1) and (x2, y2) are connected by a straight line on a curve y = f(x), the slope is given by the ratio of the y-coordinate difference to x-coordinate difference.
How to find the equation of a line from two points?
Two-point form is used to find the equation of a line passing through two points. Its formula is given by,
y – y1 = m (x – x1)
or
where,
m is the slope of line,
(x1, y1) and (x2, y2) are the two points through which line passes,
(x, y) is an arbitrary point on the line.
Derivation
Consider a line with two fixed points B (x1, y1) and C (x2, y2). Another point A (x, y) is an arbitrary point on the line.
As the points A, B and C are concurrent the slope of AC must be equal to BC.
Using the formula for slope we get,
(y – y1) / (x – x1) = (y2 – y1) / (x2 – x1)
Multiplying both sides by (x – x1) we get,
This derives the formula for two point form of a line.
Sample Problems
Problem 1. Find the equation of a line passing through the points (2, 4) and (-1, 2).
Solution:
We have,
(x1, y1) = (2, 4)
(x2, y2) = (-1, 2)
Find the slope of the line.
m = (2 – 4)/(-1 – 2)
= -2/-3
= 2/3
Using the two point form we get,
y – y1 = m (x – x1)
y – 4 = 2/3 (x – 2)
3y – 12 = 2 (x – 2)
3y – 12 = 2x – 4
2x – 3y + 8 = 0
Problem 2. Find the equation of a line passing through the points (4, 5) and (3, 1).
Solution:
We have,
(x1, y1) = (4, 5)
(x2, y2) = (3, 1)
Find the slope of the line.
m = (1 – 5)/(3 – 4)
= -4/-1
= 4
Using the two point form we get,
y – y1 = m (x – x1)
y – 5 = 4 (x – 4)
y – 5 = 4x – 16
4x – y – 11 = 0
Problem 3. Find the equation of a line passing through the points (2, 1) and (4, 0).
Solution:
We have,
(x1, y1) = (2, 1)
(x2, y2) = (4, 0)
Find the slope of the line.
m = (0 – 1)/(4 – 2)
= -1/2
Using the two point form we get,
y – y1 = m (x – x1)
y – 1 = (-1/2) (x – 2)
2y – 2 = 2 – x
x + 2y – 4 = 0
Problem 4. Find the y-intercept of the equation of a line passing through the points (3, 5) and (8, 7).
Solution:
We have,
(x1, y1) = (3, 5)
(x2, y2) = (8, 7)
Find the slope of the line.
m = (7 – 5)/(8 – 3)
= 2/5
Using the two point form we get,
y – y1 = m (x – x1)
y – 5 = (2/5) (x – 3)
5y – 25 = 2x – 6
2x – 5y + 19 = 0
Put x = 0 to get the y-intercept.
=> 2 (0) – 5y + 19 = 0
=> 5y = 19
=> y = 19/5
Problem 5. Find the x-intercept of the equation of a line passing through the points (4, 8) and (1, 3).
Solution:
We have,
(x1, y1) = (4, 8)
(x2, y2) = (1, 3)
Find the slope of the line.
m = (3 – 8)/(1 – 4)
= -5/-3
= 5/3
Using the two point form we get,
y – y1 = m (x – x1)
y – 8 = (5/3) (x – 4)
3y – 24 = 5x – 20
5x – 3y + 4 = 0
Put y = 0 to get the x-intercept.
=> 5x – 3 (0) + 4 = 0
=> 5x + 4 = 0
=> x = -4/5
Problem 6. Find the slope of a line passing through the points (2, 7) and (-4, 5).
Solution:
We have,
(x, y) = (2, 7)
(x1, y1) = (-4, 5)
Using the formula we get,
y – y1 = m (x – x1)
=> 7 – 5 = m (2 – (-4))
=> 2 = m (2 + 4)
=> 6m = 2
=> m = 1/3
Problem 7. Find the slope of a line passing through the points (4, -5) and (6, 7).
Solution:
We have,
(x, y) = (4, -5)
(x1, y1) = (6, 7)
Using the formula we get,
y – y1 = m (x – x1)
=> -5 – 7 = m (4 – 6)
=> -12 = m (-2)
=> -2m = -12
=> m = 6