We will learn the transformation of general form into intercept form.
To reduce the general equation Ax + By + C = 0 into intercept form (\(\frac{x}{A}\) + \(\frac{y}{A}\) = 1):
We have the general equation Ax + By + C = 0.
If A ≠ 0, B ≠ 0, C ≠ 0 then from the given equation we get,
Ax + By =  C (Subtracting c from both sides)
⇒ \(\frac{Ax}{C}\) + \(\frac{By}{C}\) = \(\frac{C}{C}\), (Dividing both sides by C)
⇒ \(\frac{Ax}{C}\) + \(\frac{By}{C}\) = 1
⇒ \(\frac{x}{\frac{C}{A}}\) + \(\frac{y}{\frac{C}{B}}\) = 1, which is the required intercept form (\(\frac{x}{A}\) + \(\frac{y}{A}\) = 1) of the general form of line Ax + By + C = 0.
Thus,
for the straight line Ax + By + C = 0,
Intercept on xaxis = (\(\frac{C}{A}\)) =  \(\frac{\textrm{Constant term}}{\textrm{Coefficient of x}}\)
Intercept on yaxis = (\(\frac{C}{B}\)) =  \(\frac{\textrm{Constant term}}{\textrm{Coefficient of y}}\)
Note: From the above discussion we conclude that the intercepts made by a straight line with the coordinate axes can be determined by transforming its equation to intercept form. To determine the intercepts on the coordinate axes we can also use the following method:
To find the intercept on xaxis (i.e., xintercept), put y = 0 in the given equation of the straight line line and find the value of x. Similarly To find the intercept on yaxis (i.e., yintercept), put x = 0 in the given equation of the straight line and find the value of y.
Solved examples on transformation of general equation into intercept form:
1. Transform the equation of the straight line 3x + 2y  18 = 0 to intercept form and find its xintercept and yintercept.
Solution:
The given equation of the straight line 3x + 2y  18 = 0
First add 18 on both sides.
⇒ 3x + 2y =18
Now divide both sides by 18
⇒ \(\frac{3x}{18}\) + \(\frac{2y}{18}\) = \(\frac{18}{18}\)
⇒ \(\frac{x}{6}\) + \(\frac{y}{9}\) = 1,
which is the required intercept form of the given straight line 3x + 2y  18 = 0.
Therefore, xintercept = 6 and yintercept = 9.
2. Reduce the equation 5x + 4y = 8 into intercept form and find its intercepts.
Solution:
The given equation of the straight line 7x + 4y = 8.
First divide both sides by 8
⇒ \(\frac{7x}{8}\) + \(\frac{4y}{8}\) = \(\frac{8x}{8}\)
⇒ \(\frac{7x}{8}\) + \(\frac{y}{2}\) = 1
⇒ \(\frac{x}{\frac{8}{7}}\) + \(\frac{y}{2}\) = 1,
which is the required intercept form of the given straight line 5x + 4y = 8.
Therefore, xintercept = \(\frac{8}{7}\) and yintercept = 2.
● The Straight Line
11 and 12 Grade Math
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