We will learn how to convert the general form to the intercept form.

To reduce the general equation ax + by + c = 0 into the 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 we get from the given equation

ax + by = – c (subtract c from both sides)

⇒ ( frac {ax} {- c} ) + ( frac {by} {- c} ) = ( frac {-c} {- c} ), (both sides divide by – c)

⇒ ( frac {ax} {- c} ) + ( frac {by} {- c} ) = 1

⇒ ( frac {x} {- frac {c} {a}} ) + ( frac {y} {- frac {c} {b}} ) = 1, that is the required intercept Form ( ( frac {x} {a} ) + ( frac {y} {b} ) = 1) of the general form of the line ax + by + c = 0.


Thus for the straight line ax + by + c = 0,

Axis intercept on the x-axis = – ( ( frac {c} {a} )) = – ( frac { textrm {constant term}} { textrm {coefficient of x}} )

Intercept on the y-axis = – ( ( frac {c} {b} )) = – ( frac { textrm {constant term}} { textrm {coefficient of y}} )

Note: From the discussion above, we conclude that the intercepts of a straight line with the coordinate axes can be determined by converting their equation to intercept form. To determine the intercepts on the coordinate axes, we can also use the following method:

To find the intercept on the x-axis (i.e., x-intercept), plug y = 0 into the given equation of the straight line and find the value of x. Similarly, to find the intercept on the y-axis (i.e., y-intercept), plug x = 0 into the given equation of the straight lines and find the value of y.

Solved examples for transforming the general equation into the intercept form:

1. Transform the equation of the straight line 3x + 2y – 18 = 0 into the sectional form and determine its x-axis intercept and y-axis 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,

this is the required form of intersection of the given straight line 3x + 2y – 18 = 0.

Hence x-intercept = 6 and y-intercept = 9.

2. Reduce the equation -5x + 4y = 8 to the 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,

this is the required form of intersection of the given straight line -5x + 4y = 8.

Hence x-intercept = ( frac {8} {7} ) and y-intercept = -2.

The straight line

11th and 12th grade math degree

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