# INTERCEPT FORM EQUATION OF A LINE

## About "Intercept form equation of a line"

"Intercept form equation of a line" is one of the ways to express equation of a line.

Here, we will form equation of a straight line using x- intercept and y - intercept.

Intercept form equation of a line

Here,

x- intercept  =  a

y- intercept  =  b

## Intercept form equation of a line - Practice problems

Problem 1 :

If the x-intercept and y-intercept of a straight line are 2/3 and 3/4 respectively, find the general equation of the straight line.

Solution :

Given : x- intercept  "a"  =  2/3 and y-intercept  "b"  =  3/4

So, the equation of the straight line in intercept form is

x/a  +  y/b  =  1

Plugging  a  =  2/3  and  b  =  3/4, we get

x / (2/3)  +  y / (3/4)  =  1

3x / 2  +  4y / 3  =  1

(9x + 8y)  / 6  =  1

9x + 8y  =  6 -------> 9x + 8y - 6  =  0

Hence the general equation of straight line is 9x + 8y - 6 = 0

Problem 2 :

Find the equations of the straight lines each passing through the point (6, -2) and whose sum of the intercepts is 5.

Solution :

Let "a" and "b" be the x-intercept and y-intercept of the required straight line respectively.

Given : Sum of the intercepts = 5

So, we have  a + b  =  5 --------> b = 5 - a

Now, equation of the straight line in intercept form is

x/a  +  y/b  =  1

Plugging  b  =  5 - a, we get

x / a  +  y / (5-a)  =  1

[(5-a)x + ay ] / a(5-a) = 1

(5-a)x + ay  =  a(5-a) ----------(1)

Since the straight line (1) is passing through (6, -2),

we can plug x = 6 and y = -2 in (1)

(1) --------> (5 - a)6 - 2a  =  a(5 - a)

30 - 6a - 2a  =  5a - a² ---------> a² - 13a + 30 = 0

a² - 13a + 30 = 0 -----------> (a - 10)(a - 3) = 0

a  =  10 and a  =  3

When a = 10, (1)------->(5 - 10)x + 10y  =  10(5 - 10)

- 5x + 10y  =  - 50 -------> 5x - 10y  - 50  =  0

5x - 10y - 50  =  0 -------> x - 2y - 10  =  0

When a = 3, (1)------->(5 - 3)x + 3y  =  3(5 - 3)

2x + 3y  =  6 --------> 2x + 3y - 6  =  0

Hence, x - 2y - 10 = 0 and 2x + 3y - 6 = 0 are the general equations of the required straight lines

Problem 3 :

Find the x - intercept and y - intercept of the straight line whose equation is 5x + 3y - 15  =  0.

Solution :

To find the x - intercept and y - intercept of the line 5x + 3y -15 = 0, first we have to write the given equation in intercept form.

So, let us write the given equation in intercept form.

5x + 3y - 15  =  0

5x + 3y  =  15

Divide by 15 on both sides,

(5x/15) + (3y/15)  =  15/15

x/3 + y/5  =  1

The above equation is in intercept form.

Hence, x- intercept is 3 and y - intercept is 5.

## Other forms of equation of a straight line

Apart from intercept form equation of a straight line, we have some other different forms of equation of a straight line.

They are

(i)  Slope intercept form

(ii) Point slope form

(iii) Two points form.

Let us look at the above different forms of equation of a straight line in detail.

## Slope intercept form

Here,

Slope of the line  =  m

y-intercept  =  b

## Point slope form

Here,

Slope of the line  =  m

Point  =  (x₁ , y)

## Two point form

Here, the two points are

(x₁ , y) and (x₂ , y)

Apart from the above forms of equation of straight line, there are some other ways to get equation of a straight line.

1. If a straight line is passing through a point (0,k) on y-axis and parallel to x-axis, then  the equation of the straight line is   y = k

2. If a straight line is passing through a point (c,0) on x-axis and parallel to y-axis, then  the equation of the straight line is   x = c

3. Equation of x-axis is  y  =  0.

(Because, the value of "y" in all the points  on x-axis is zero)

4. Equation of y-axis is  x  =  0.

(Because, the value of "x" in all the points  on y-axis is zero)

5. General equation of a straight line is

ax + by + c  =  0

After having gone through above stuff, we hope that students would have understood the stuff "Intercept form equation of a line".

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