Intercept

In Mathematics, an intercept is a point on the y-axis whereby the slope of a line passes. It is the y-coordinate of a point on the y-axis where a straight line or a curve intersects it. This is what we get when we put in the equation for a line, y = mx+c, where m is the slope and c is the y-intercept.

The two types of intercepts are the x-intercept and the y-intercept. The x-intercept is the actual point on the x-axis where the line crosses it, while the y-intercept is the point where the line crosses the y-axis. We will explain what the intercept is, how to get the intercept for a given line, and how to graph intercepts with examples in this article. So, let us start by understanding the meaning of intercept in the coming section.

What is the Meaning of Intercept?

A line is a one-dimensional geometric shape. It is measured for 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 the 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 the X-axis gives the x-intercept definition and the point of intersection of the line with the Y-axis gives the y-intercept definition in Maths. If the axis is not mentioned, then it generally represents the y-intercept definition in Maths and is given by the formula:

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’ into either side, 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 of Equation of a Line

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Consider a line intersecting the X and Y axes at 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:

\[\frac {y-y_1}{y_2 - y_1} = \frac {x - x_1}{x_2 -x_1}\]

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-intercepts 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 = \[\frac {12}{4}\]

y= 3

To find the ‘x’ intercept, the value of the 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 = \[\frac {12}{3}\]

x = 4

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

2. Find the x and y-intercepts of the line represented by 20y = 10 – 40x.

Solution:

The x-intercept is of the form (s, 0).

Let us put y = 0 in the equation of the given line:

20*(0) = 10 – 40x 

or, 0 + 40x = 10 

or, x = \[\frac {1}{4}\]

Thus, the x-intercept of the given line is \[\frac {1}{4}\] or 0.25.

The y-intercept is of the form (0, t).

Let us put x = 0 in the equation of the given line:

20y = 10 – 40*(0)

or, 20y = 10

or, y = \[\frac {1}{4}\]

Thus, the y-intercept of a given line is \[\frac {1}{2}\] or 0.5.

3. Intersect the x-axis and y-axis with two intercepts P(2,0) and Q(0,3), respectively. Try to determine the equation of the line.

Solution: Two intercepts, P(2,0) and Q(0,3), cross the x-axis and y-axis, respectively.

We may derive the following from the line's equation,

\[\frac {x}{a}\] + \[\frac {y}{b}\] = 1 ……….. (1)

Here, a = 2 and b = 3

As a result, when the values of intercepts a and b are put in equation 1, we get:

→ \[\frac {x}{2}\] + \[\frac {y}{3}\] = 1

→ 3x + 2y = 6

→ 3x + 2y – 6 = 0,

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

Fun Facts about Intercept Meaning in Maths

• At the point of intersection of the line with the 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 the 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 Maths.

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