We will learn the transformation of general form into slope-intercept form.
To reduce the general equation Ax + By + C = 0 into slope-intercept 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 slope-intercept 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 slope-intercept form:
1. Transform the equation of the straight line 2x + 3y - 9 = 0 to slope intercept form and find its slope and y-intercept.
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 slope-intercept form of the given straight line 2x + 3y - 9 = 0.
Therefore, slope of the given line (m) = -\(\frac{2}{3}\) and y-intercept = 3.
2. Reduce the equation -5x + 2y = 7 into slope intercept form and find its slope and y-intercept.
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 slope-intercept form of the given straight -5x + 2y = 7.
Therefore, slope of the given straight line \(\frac{5}{2}\) and y-intercept \(\frac{7}{2}\).
● The Straight Line
11 and 12 Grade Math
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