# General Form into Intercept Form

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

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