We will learn the transformation of general form into intercept form.
To reduce the general equation ax + by + c = 0 into intercept form (\(\frac{x}{a}\) + \(\frac{y}{b}\) = 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 (\(\frac{x}{a}\) + \(\frac{y}{b}\) = 1) of the general form of line ax + by + c = 0.
Thus,
for the straight line ax + by + c = 0,
Intercept on x-axis = -(\(\frac{c}{a}\)) = - \(\frac{\textrm{Constant term}}{\textrm{Coefficient of x}}\)
Intercept on y-axis = -(\(\frac{c}{b}\)) = - \(\frac{\textrm{Constant term}}{\textrm{Coefficient of y}}\)
Note: From the above discussion we conclude that the intercepts made by a straight line with the co-ordinate axes can be determined by transforming its equation to intercept form. To determine the intercepts on the co-ordinate axes we can also use the following method:
To find the intercept on x-axis (i.e., x-intercept), put y = 0 in the given equation of the straight line line and find the value of x. Similarly To find the intercept on y-axis (i.e., y-intercept), 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 x-intercept and y-intercept.
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, x-intercept = 6 and y-intercept = 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, x-intercept = \(\frac{8}{7}\) and y-intercept = -2.
● The Straight Line
11 and 12 Grade Math
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