Intercept

What is the Meaning of Intercept?

A line is a one dimensional geometric shape. It is measured with respect to only one dimension (length). A line can generally be represented as a planar figure in a coordinate plane. The coordinate plane has two perpendicular axes called the horizontal ‘X’ axis and vertical ‘Y’ axis in the cartesian coordinate system. The points where the line crosses the two axes are called the intercepts. The point of intersection of the line with X axis gives the x intercept definition and the point of intersection of the line with Y axis gives the y intercept definition in Math. If the axis is not mentioned then it generally represents the y intercept definition in Math and is given by the formula:

B = y - Ax


In the above equation, B represents the y intercept meaning and A represents the slope of the line.

What is Intercept?

The general equation of a line is given as:

Ax + By + C = 0 →(1)

In the above equation, A, B and C may be any real numbers.

If the above equation is divided by ‘B’ on either sides, we get

\[\frac{A}{B}\]x + \[\frac{B}{B}\]y + \[\frac{C}{B}\] = 0

\[\frac{A}{B}\]x + y + \[\frac{C}{B}\] = 0

y = \[\frac{-A}{B}\]x + \[\frac{-C}{B}\]

In the above equation, (-A/B) represents the slope of the line and is represented as ‘m’ and (-C/B) is the y intercept of the line and is represented as ‘c’. So, the equation of the line becomes

y = mx + c

The intercept of a line can also be found by substituting either ordinate or abscissa as zero in the equation for the line. If y intercept is to be found, then the value of x coordinate is substituted as zero and if the x intercept is to be found, the value of y coordinate is substituted as zero in the general equation for the line. 

If a line is intersecting the x and y axes at ‘a’ and ‘b’ respectively, then the equation of a line can also be written as:

\[\frac{x}{a}\] + \[\frac{y}{b}\] = 1

Proof for Intercept form Equation of a Line:

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Consider a line intersecting the X and Y axes at the points a and b as shown in the figure above. The coordinates of the point of intersection of the line with the X axis is given as P (x1, y1) = (a, 0). Similarly, the point of intersection of the line with the Y axis is given as Q (x2, y2) = (0, b). The equation of a line whose two points are given is as follows:

\[\frac{y-y1}{y2-y1}\] = \[\frac{x-x1}{x2-x1}\]

Substituting the values of x1, x2, y1 and y2 in the above equation, we get

\[\frac{y-0}{b-0}\] = \[\frac{x-a}{0-a}\]

\[\frac{y}{b}\] = \[\frac{x-a}{-a}\] 

\[\frac{y}{b}\] = - \[\frac{x}{a}\] + \[\frac{-a}{-a}\]

\[\frac{y}{b}\] + \[\frac{x}{a}\] = 1

The above equation gives the equation of a line in the form of its intercept. 

Intercept Example Problems:

1. Find the x and y Intercept of the line Represented by 3x + 4y = 12.

Solution:

To find the ‘y’ intercept of the line, the value of x should be taken as 0 in the equation for the line. The equation of the line is given as:

3x + 4y = 12

3(0) + 4y = 12

4y = 12

y = \[\frac{12}{4}\] = 3

To find the ‘x’ intercept, the value of y coordinate should be 0 in the equation for the line. X intercept is calculated as follows.

3x + 4y = 12

3x + 4(0) = 12

3x = 12

x = \[\frac{12}{3}\] = 4

The x and y intercepts of the line are 4 and 3 respectively.

Fun Facts about Intercept Meaning in Maths:

  • At the point of intersection of the line with X axis, the y coordinate is zero. So the X intercept is found by substituting y = 0 in the equation for the line.

  • At the point of intersection of a line with Y axis, the x coordinate is zero. So, the y coordinate can be found as the value of y at the point (0, y) on the line. This gives the y intercept definition in Math.

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FAQ (Frequently Asked Questions)

1. What is Intercept? How is it Determined?

Intercept meaning in Maths can be explained with the standard equation of a line in the cartesian coordinate system. An intercept is the point of intersection of the line with the coordinate axes. If the line intersects with the X  axis, then the y coordinate of the point of intersection is zero and the intercept is called the x intercept. Similarly, at the point of intersection of the line with the Y axis, the x coordinate is zero and the y coordinate of the point gives the y intercept of the line. 

2. What is the General Representation of a Line?

The general equation of a line is Ax + By + C = 0. Here, A, B and C are any real numbers. ‘x’ and ‘y’ are the variables representing any point on the line in the cartesian coordinate system. This general equation is most commonly used in the form of slope - intercept equation. In this form, the line is represented in terms of its slope and intercept. The general equation of a line is divided by ‘B’ on both the sides. So, the equation becomes (A/B) x + (B/B) y + (C/B) = (0/B).

In the above equation, (-A/B) represents the slope of the line and (-C/B) gives the y intercept meaning. If the slope is represented as ‘m’ and y intercept meaning in Maths as ‘c’, the equation of the line becomes 

y = mx + c