General Equation of a Line
general equation of a line:
AX + BY+C = 0 is the general equation of the line, where A, B both cannot be zero.
A is the co efficient of X
B is the co efficient of Y
C is the constant.
An equation of first degree is always a straight line.
In the graph, we can see three lines.
equation of the horizontal line is Y=4
equation of the blue line is Y = 3x+2 or 3x -y +2 =0
equation of the green line is Y = -x+2 or x+y -2 =0
for all these equations, degree is 1.
We can also note that Constant (c)=0 for the lines that pass through origin.
As we learn about lines, we should also the know the concept of slope.
what is a slope?
The slope is the angle which the line makes with the X axis. In other words, slope determines the steepness of the line.
The formula for slope:
If two points on the line are given, slope (m) = (y2 - y1) / (x2 - x1)
If the general equation Ax + By +C =0 is given, then slope (m) = (-coefficient of x)/ (coefficient of y)
The line Y = 4 is parallel to X axis, it does not make an angle with X - axis. Slope (m) = 0.
Similarly the line X =4 would be parallel to Y axis, it makes an angle of 900 with the X axis, slope = ∞
The slope of the line 3x -y +2 = 0 is slope (m) = (-coefficient of x)/ (coefficient of y) = (-3)/(-1) = 3
The slope of the line x+y -2 = 0 is slope (m) = (-coefficient of x)/ (coefficient of y) = (-1)/1 = -1
we can find the equation of the line:
a) If two points on the line are given.
b) If one point on the line and the slope is given.
Find the equation of the line which passes through the two points (4,3) (1,4)
solution:
`(y-y`1)/ (y2-y1) = (x-x1)/ (x2-x1) where (x1,y1) (x2,y2) are two given points.
The two given points are (4,3) (1,4)
(y - 3)/ (4 - 3) = (x - 4)/ (1-4)
y - 3 = (-1/3)(x - 4)
3y -9 = -x +4
3y +x -13 = 0 is the general equation of the line.
Find the equation of the line whose slope is 1/2 and it passes through the point (3,4)
Solution:
(y-y1) = m (x-x1) where m is the slope and (x1,y1) is the given point
y - 4 = 1/2 (x - 3)
2y - 8 = x -3
2y -x - 5 = 0 is the general equation of the line.
