Today is our topic of discussion Equation of Straight Lines .
Equation of Straight Lines
Suppose, a definite straight line L passes through two definite points A(3, 4) and B(5,7). In the figure below, the line is shown.
Then the slope of the straight line AB is
m= (7 – 4)/(5 – 3) = 3/2 ………………..(1)
Suppose, P(x, y) is any point on the line L. Then the slope of the line AP is m₂ = y-4 /x-3…………… (2)
Since AB and AP are segments of the same line, both have the same slope. i.e. m₁ = m2
or, 3/2 = (y – 4)/(x – 3) [From (1) and (2)]
or, 3x – 9 = 2y – 8
or, 2y = 3x – 1
or, y = 3/2x – 1/2 (3)
again, slope of PB, m3 = 7 – y/5…………….. (4)
As slopes of AB and PB are same,
m₁ = m3
or, 3/2= 7-y/5-x [From (1) and (4)]
or, 15-3x=14 – 2y
or, 2y+15=3x+14
or, 2y=3x+1
or, y= 3/2x – 1/2……………….. (5)
The equation (3) and (5) is the same equation. So, the equation (3) or (5) is the Cartesian equation of the straight line L. If we observe, we will find that the equation (3) or (5) is the single equation of x and y and it indicates a straight line. So, undoubtedly it can be said that the single equation of x and y always indicates a straight line. The equations (3) and (5) can be expressed in the following way:
and the Cartesian equation of that straight line will be:
y-y1/x-x1 = m… (6), y – Y2/x-x2 = m…… (7)
From equation (6):
y-y₁ = m(x-x1)… (8)
From equation (7):
y-y2m(x-x2)… (9)
From (8) and (9) we can say, if the slope of the line is m and the line passes through the definite points (1, 1) and (22, y2), the Cartesian equation of the line will be determined by the equation (8) or (9). From the equation (6) and (7) we get,
m = y-y1/x-x1= y-y2/x-x2 (10)
From the equation (10), it is said clearly, if a straight line passes through two definite points A(x1,y1) and B(T2, y2), its Cartesian equation will be
The above discussion is explained with the help of the following examples so that the students can easily understand the slope of the straight line and the equation.
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