The equation of a straight line is satisfied by the co-ordinates of every point lying on the straight line and not by any other point outside the straight line.
Equation of a line using two points on the line :
(y - y_{1})/(y_{2} - y_{1}) = (x - x_{1})/(x_{2} - x_{1})
Here (x_{1}, y_{1}) and (x_{2}, y_{2}) are the points on the line.
Example 1 :
Find the equation of the line which is passing through the points (2, -8) and (-5, 2).
Solution :
Here (x_{1}, y_{1}) = (2, -8) and (x_{2}, y_{2}) = (-5, 2).
Equation of a line :
(y - y_{1})/(y_{2} - y_{1}) = (x - x_{1})/(x_{2} - x_{1})
(y - (-8))/(2 - (-8)) = (x - 2)/(-5 - 2)
(y + 8)/(2 + 8) = (x - 2)/(-7)
(y + 8)/10 = (x - 2)/(-7)
-7(y + 8) = 10(x - 2)
-7y - 56 = 10 x - 20
10 x + 7y - 20 + 56 = 0
10 x + 7y + 36 = 0
Example 2 :
Find the equation of the line which is passing through the points (3, 0) and (4, -1).
Solution :
Here (x_{1}, y_{1}) = (3, 0) and (x_{2}, y_{2}) = (4, -1).
Equation of a line :
(y - 0)/(-1 - 0) = (x - 3)/(4 - 3)
y/(-1) = (x - 3)/1
1(y) = -1(x - 3)
y = - x + 3
x + y - 3 = 0
Example 3 :
Find the equation of the line which is passing through the points (-1, -5) and (-3, -1).
Solution :
Here (x_{1}, y_{1}) = (-1, -5) and (x_{2}, y_{2}) = (-3, -1).
Equation of a line :
(y - (-5))/(-1 - (-5)) = (x - (-1))/(-3 - (-1))
(y + 5)/(-6) = (x + 1)/(-3 + 1)
(y + 5)/(-6) = (x + 1)/(-2)
-2(y + 5) = -6(x + 1)
-2y - 10 = -6 x - 6
6 x - 2 y - 10 + 6 = 0
6 x - 2 y - 4 = 0
÷ by 2 => 3 x - y - 2 = 0
Example 4 :
Find the equation of the line which is passing through the points (0,-5) and (4,-6).
Solution :
Here (x_{1}, y_{1}) = (0, -5) and (x_{2}, y_{2}) = (4, -6).
Equation of a line :
(y - (-5))/(-6 - (-5)) = (x - 0)/(4 - 0)
(y + 5)/(-6 + 5) = x/4
(y + 5)/(-1) = x/4
4(y + 5) = -1(x)
4y + 20 = - 1 x
x + 4y + 20 = 0
Example 5 :
Find the equation of the line which is passing through the points (4, -3) and (0, -2).
Solution :
Here (x_{1}, y_{1}) = (4, -3) and (x_{2}, y_{2}) = (0, -2).
Equation of a line :
(y - (-3))/(-2 - (-3)) = (x - 4)/(0-4)
(y + 3)/(-2 + 3) = (x - 4)/(-4)
(y + 3)/1 = (x - 4)/(-4)
-4(y + 3) = 1(x - 4)
-4y - 12 = 1 x - 4
x + 4y + 12 - 4 = 0
x + 4y + 8 = 0
Example 6 :
Find the equation of the line which is passing through the points (2, -1) and (-1, -7).
Solution :
Here (x_{1}, y_{1}) = (2, -1) and (x_{2}, y_{2}) = (-1, -7).
Equation of a line :
(y - (-1))/(-7 - (-1)) = (x - 2)/(-1 - 2)
(y + 1)/(-7 + 1) = (x - 2)/(-3)
(y + 1)/(-6) = (x - 2)/(-3)
-3(y + 1) = -6(x - 2)
-3y - 3 = -6 x + 12
6 x - 3 y - 3 - 12 = 0
6 x - 3 y - 15 = 0
÷ by 3 => 2 x - y - 5 = 0
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