Intercept Form

The point where a line or a curve intersects or crosses the graph’s coordinate axis is an intercept. The point on the line that crosses or intersects the x-axis is the x-intercept. And the point on the line that crosses or intersects the y-axis is called the y-intercept. Usually, when the axis is not specified, we take the y-axis. It is denoted by the letter “b”.

Overview of intercept form

Y = mx + c is the line equation that meets the y-axis at a point (0,c).

To write the line’s intercept form, replace c with b. As a result, the equation is:

y = mx + b

As a result, the formula for the y-intercept of a line is:

y – mx = b, where b represents the intercept.

m is the line’s slope.

y and x are the points on the y-axis and the x-axis, respectively.

How do you convert the intercept form of a line to the standard form of a line?

The intercept form of a line of an equation is changed into its standard form through a basic formula. The equation of this intercept form is x/a + y/b = 1. 

x/a + y/b = 1

(bx + ay)/ab = 1

bx + ay = ab

The derived standard form of the equation of a line is ax + by + c = 0.

Identifying the X and Y Intercepts:

Consider the following equation in the form of a straight line:

C =A X + By.

Divide the equation by C

C/C = (Ax/C) + (By/C)

1 = [x/(C/A)] + [y/(C/B)]

When comparing this equation to the intercept form equation of a line, (x/a) + (y/b) = 1.

As a result, we get – x-intercept = a = C/A.

B/C = y-intercept = b

Alternatively,

Substitute y = 0 and solve for x to find the x-intercept.

X + B(0) = C, for example.

C = X

C/A = x

Substitute x = 0 and solve for y to find the y-intercept.

A(0) + By = C, for example.

C = by

C/B = y

Graph of intercept form:

The intercepts of a graph are the locations where the two axes, the x-axis, and the y-axis cross. Consider the case when two intercepts, P (5,0) and Q (0,4), connect the x- and y-axes, respectively.

 Determine the equation of the line. P (5,0) and Q (0,4) are two intercepts that cross the x- and y-axes.

We know the line’s equation.

1 = x/a + y/b………. (1)

In this situation, a = 5 and b = 4.

As a result, when the values of intercepts a and b are inserted into equation 1, the following results are obtained:

=>x/5 + y/4 = 1

=> 4x + 5y = 20

=> 4x + 5y – 20 = 0

Numerical Examples of intercept form:

• Find X and Y-intercept for a given straight line as 5x + 2y = 10.

Follow these steps to locate the x-intercept:

Assume the equation Ax + By = C, which is a line.

To find the x-intercept, substitute y = 0 and solve for x.

The y-intercept can be found by substituting x = 0 and solving for y.

 The equation 5x + 2y = 10 is a straight line.

To locate the x-intercept.

Substitute y = 0 in the given equation.

10 = 5x + 2(0)

5x=10 

2 = x

To find the y-intercept, 

Substitute x = 0 for x in the above equation.

10 = 5(0) + 2y

2y = 10

5 = y

As a result, the x-intercept is (2, 0), and the y-intercept is (0, 5)

• Find the line formula with –3 and 2 as the x and y-axis intercepts, respectively.

 Ans. Given, a = –3 , b = 2.

By intercept form, we know that:

x/a + y/b = 1

x/-3 + y/2 = 1

Or

2x – 3y + 6 = 0.

Hence, this is the required equation.

• Find the equation of line having the centre of the intercepts at x and y-axis as (1,2).

Ans. The equation for a line that makes intercepts a and b with the x- and y-axes, respectively, is:

1 = x/a + y/b

(a +0)/2 = 1 ⇒ a=2

(b +0)/2=2⇒b=4

As a result, the line’s required equation is;

1 = x/2 + y/4

0 = 2x + y – 4

Conclusion:

Intercept is the point where a curve or line intersects the graph’s axis. The y-intercept is a point that intersects the y-axis; y = mx + c is the equation of the line that meets the y-axis at a point. For the y-intercept of a line, it is:

y – mx = b

Intercept Form of Equation of a Line:
x/a + y/b = 1.

The intercepts of a graph are the locations where the two axes, x and y, cross.