An intercept is a point on y-axis, through which the slope of the line passes. It is the value of y-axis, where the line is crossing the axis. This is represented when we write the equation for a line. In this article, we are going to discuss the intercept definition in detail.

Also read: Equation of plane in intercept form

Intercept Definition

The point where the line or curve crosses the axis of the graph is called intercept. If a point crosses the x-axis, then it is called x-intercept. If a point crosses the y-axis, then it is called y-intercept.

The intercept of a line is the point at which it intersects either the x-axis or y-axis. If the axis is not specified, usually the y-axis is considered. It is normally denoted by the letter ‘b’.

Except that line is accurately vertical, it will constantly cross the y-axis somewhere, even if it is way off the top or bottom of the chart.

Intercept Formula

The equation of the line, which intersects the y-axis at a point is given by:
                            y = mx + c
Now, we have to write the intercept form of the line, we can replace c with b. Thus, the equation becomes:

                           y = mx + b
Hence, the formula for the intercept of a line is given by:
                          b = y – mx
Where, b is the intercept, m is the slope of the line and y and x indicate the points on y-axis and x-axis respectively.

Now, another way of writing the equation of the line, considering a line is intersecting the x-axis and y-axis at point a and b respectively.
                        x/a + y/b = 1
Here, a and b are the intercepts of the line which intersect the x-axis and y-axis, respectively. The values of a and b can be positive, negative or zero and explain the position of the points at which the line cuts both axes, relative to the origin.

How to Find X and Y Intercepts?

Consider a striaght line equation Ax + By = C.

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

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

Example: Let us assume the straight-line equation 5x +2y =10

To find x-intercept: 

Substitiute y=0 in the given equation

5x + 2(0) = 10

5x =10

x =2

To find y-intercept

Substitiute x =0 in the given equation

5(0) + 2y =10

2y = 10

y = 5

Therefore, x -intercept is (2, 0)

y -intercept is (0, 5)

Intercept Form Proof

The formula of the line formed by the two points is given by:

y-y1/y2-y1 = x-x1/x2-x1

Intercept Form

Say, P(a, 0) = (x1, y1) and Q(0, b) = (x2, y2) are the two points of the line which cuts the x-axis and y-axis, relative to the origin(0,0). Then the formula becomes:

=> y – 0 / b – 0 = x – a/ 0 – a

=> y/b = x/-a – a/-a

=> x/a + y/b = 1

Hence, proved.

Intercept Example

Let two intercepts P(2,0) and Q(0,3) intersect the x-axis and y-axis, respectively. Find the equation of the line.

Solution: Given, two intercepts P(2,0) and Q(0,3) intersect the x-axis and y-axis.

From the equation of the line we know,

x/a + y/b = 1 ……….. (1)

Here, a = 2 and b = 3

Therefore, putting the values of intercepts a and b, in equation 1, we get:

=>x/2 + y/3 = 1

=> 3x + 2y = 6

=> 3x + 2y – 6 = 0,

Therefore, the equation of the line is 3x + 2y – 6 = 0.

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