What is an Intercept and How to Find Intercepts?

Intercepts are the point at which the graph intersects the axes. Usually, we work in 2-dimension which means that there are two axes: the x-axis (abscissa) and y-axis (ordinate). Since intercepts are related to axes, there are two types of intercepts: x-intercept and y-intercept. If the graph crosses the x-axis, the graph's point intersects the x-axis is the x-intercept. If the graph intersects the y-axis, the graph's point intersects the y-axis is called the y-intercept. For better understanding, check the below graph:

The graph intersects both axes but which one is the x-intercept and which one is the y-intercept. The point, (a, 0) is the x-intercept and the point (0, b) is the y-intercept. Did you notice something? The coordinates of the intercepts have their opposite axis equal to zero. Consider the x-intercept, (a, 0), the y-axis is zero and the same thing is happening in the y-intercept as well. Is this an incident? No, the opposite axis should always be zero when finding the intercept. In simple words, the y-axis in the x-intercept should always be zero and the x-axis in the y-intercept should always be zero.

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

a is the x-intercept.

b is the y-intercept.

a and b must be nonzero.

The values of a and b can be obtained from the general form equation.

If y = 0, x = a.

If x = 0, y = b.

Imagine you are given an equation of a line, y = 3x - 8, and your teacher asks you to find the x-intercept as well as the y-intercept. The solution is pretty simple, we will find the x-intercept first, but it is your choice which intercept you want to find first. For the x-intercept, we know that the y-axis will be equal to zero. Apply this condition to the equation:

y = 3x - 8 \qquad y = 0

0 = 3x - 8

x = \frac { 8 }{ 3 }

Many students make this common mistake. They finish till here but there is one thing they are missing. The question asked for the x-intercept's coordinate, therefore, end the answer with the coordinates of the respective intercept. Since we were finding the x-intercept, the final answer will be x-intercept = (\frac { 8 }{ 3 }, 0), now let's find the y-intercept.

y = 3x - 8 \qquad x = 0

y = 0 - 8

y = 8

y-intercept = (0, 8)

Some Important Points to Note

A line does not have an intercept form equation in the following cases:

1.A line parallel to the x-axis, which has the equation y = k.

2.A line parallel to the x-axis, which has the equation x = k.

3.A line that passes through the origin, which has equation y = mx.

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Examples

Example 1

A line has an x-intercept of 5 and a y-intercept of 3. Find its equation.

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

\frac { x }{ 5 } + \frac { y }{ 3 } = 1

Example 2

The line x - y + 4 = 0 forms a triangle with the axes. Determine the area of the triangle.

The line forms a right triangle with the origin and its legs are the axes.

If y = 0, x = -4 = a

If x = 0, y = 2 = b

The intercept form is:

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

\frac { x }{ -4 } + \frac { y }{ 2 } = 1

The area is:

S = \frac { 1 }{ 2 } a . b

S = \frac { 1 }{ 2 } \left | (-4) . 2 \right | = 4 { u }^{ 2 }

 

Example 3

A line passes through the point A = (1, 5) and creates a triangle of 18 { u }^{ 2 } with the axes. Determine the equation of the line.

Apply the intercept form:

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

\frac { 1 }{ a } + \frac { 5 }{ b } = 1

The area of the triangle is:

18 = \frac { 1 }{ 2 } a . b

Solve the system:

\left\{\begin{matrix} b  + 5a = ab \\ ab = 36 \end{matrix}\right

After solving simultaneously, the results are:

a = 6 \qquad b = 6

a = \frac { 6 }{ 5 } \qquad b = 30

Therefore,

\frac { x }{ 6 } + \frac { y }{ 6 } = 1 \qquad x + y = 6

\frac { x }{ \frac { 6 }{ 5 } } + \frac { y }{ 30 } = 1 \qquad 25x + y = 30

 

Example 4

A line forms a triangle with the axes where the length of the leg formed by the x-axis is twice the length of the leg formed by the y-axis. If the line passes through the point A = (3, 2), what is its equation?

\frac { 3 }{ 2b } + \frac { 2 }{ b } = 1 \qquad 3 + 4 = 2b \qquad b = \frac { 7 }{ 2 }

a = 2b \qquad a = 7

\frac { x }{ 7 } + \frac { y }{ \frac { 7 }{ 2 } } = 1 \qquad x  + 2y - 7 = 0

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.