We will learn the transformation of general form into slopeintercept form.
To reduce the general equation Ax + By + C = 0 into slopeintercept form (y = mx + b):
We have the general equation Ax + By + C = 0.
If b ≠ 0, then from the given equation we get,
By =  Ax  C (Subtracting ax from both sides)
⇒ y=  A/Bx  C/B, [Dividing both sides by b (≠0).
⇒ y = (\(\frac{A}{B}\))x + (\(\frac{C}{B}\))
Which is the required slopeintercept form (y = mx + b) of the general form of line Ax + By + C = 0, where m = \(\frac{A}{B}\), b = \(\frac{C}{B}\)
Thus, for the straight line Ax + By + C = 0,
m = slope = \(\frac{A}{B}\) =  \(\frac{\textrm{Coefficient of x}}{\textrm{Coefficient of y}}\)
Note:
To determine the slope of a line by the formula m =  \(\frac{\textrm{Coefficient of x}}{\textrm{Coefficient of y}}\) first transfer all terms in the equation on one side.
Solved examples on transformation of general equation into slopeintercept form:
1. Transform the equation of the straight line 2x + 3y  9 = 0 to slope intercept form and find its slope and yintercept.
Solution:
The given equation of the straight line 2x + 3y  9 = 0
First subtract 2x from both sides.
⇒ 3y  9 = 2x
Now add 9 on both sides
⇒ 3y = 2x + 9
Then divide both sides by 3
⇒ y = (\(\frac{2}{3}\))x + 3, which is the required slopeintercept form of the given straight line 2x + 3y  9 = 0.
Therefore, slope of the given line (m) = \(\frac{2}{3}\) and yintercept = 3.
2. Reduce the equation 5x + 2y = 7 into slope intercept form and find its slope and yintercept.
Solution:
The given equation of the straight line 5x + 2y = 7.
Now solve for y in terms of x.
⇒ 2y = 5x + 7
⇒ y = (\(\frac{5}{2}\))x + \(\frac{7}{2}\), which is the required slopeintercept form of the given straight 5x + 2y = 7.
Therefore, slope of the given straight line \(\frac{5}{2}\) and yintercept \(\frac{7}{2}\).
● The Straight Line
11 and 12 Grade Math
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