X & Y intercept Formulas, Examples - Maths - TRUNG TÂM GIA SƯ TÂM TÀI ĐỨC

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Intercept

In Maths, an intercept is a point on the y-axis, through which the slope of the line passes. It is the y-coordinate of a point where a straight line or a curve intersects the y-axis. This is represented when we write the equation for a line, y = mx+c, where m is slope and c is the y-intercept.

There are basically two intercepts, x-intercept and y-intercept. The point where the line crosses the x-axis is the x-intercept and the point where the line crosses the y-axis is the y-intercept. In this article, you will learn what is the intercept, how to find the intercept for a given line, graphing intercepts along with solved examples.

Definition of Intercept

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 the x-intercept. If a point crosses the y-axis, then it is called the y-intercept.

The meaning of 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 y-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 the 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.

X and Y Intercept

X and Y-intercept are the lengths on the x-axis and y-axis, which is the distance of the point where the line or curve cuts the respective axis, from the origin. If the line cuts the x-axis at the point(a, 0), then ‘a’ is the x-intercept, and if the line cuts the y-axis at the point (0, b), then ‘b’ is the y-intercept. X and Y-intercept are useful to find the slope, equation of a line, and also to find the area made by the line with the coordinate axes.

How to Find X and Y Intercepts?

Consider a straight line equation Ax + By = C.

Divide the equation by C,

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

Comparing this equation with the equation of a line in intercept form, (x/a) + (y/b) = 1,

We get, x-intercept = a = C/A

y-intercept = b = B/C

Alternatively,

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

i.e. Ax + B(0) = C

Ax = C

x = C/A

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

i.e. A(0) + By = C

By = C

y = C/B

Go through the example given below to understand this concept in a better way.

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

To find x-intercept:

Substitute y=0 in the given equation

5x + 2(0) = 10

5x =10

x =2

To find y-intercept

Substitute 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)

Two Point Form

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

y-y₁/y₂-y₁ = x-x₁/x₂-x₁

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.

Slope Intercept Form

The equation of the line making an intercept c on the y-axis and having slope m is given by:

y = mx + c

Note: The value of c could be positive or negative since the intercept is drawn on the positive or negative side of the y-axis, respectively.

Intercept Graph

The intercepts are the points on a graph at which the graph crosses the two axes (x-axis and y-axis). The point where the graph crosses the x-axis is called the x-coordinate and the point where the graph crosses the y-axis is called the y-coordinate.

In the above intercept graph, where a line L makes x-intercept a and y-intercept b on the axes.

Thus, the equation of the line making intercepts a and b on the x-and y-axis, respectively, is:

x/a + y/b = 1

Graph Of X And Y Intercept

X and y-intercept can be found by observing the graph of the line. By knowing the points on the axis where the graph line cuts the coordinate axis, we can find the x and y-intercept. Observe the below graph and try to find the points where the line cuts the x-axis and the y-axis.

Here, the x-intercept is the distance from the origin on the x-axis and it is -4, as the line cuts the x-axis at the point (-4,0). And the y-intercept is the distance from the origin on the y-axis and it is 6 as the line cuts the y-axis at the point (0, 6).

Uses Of X And Y Intercept

The x and y-intercept are useful to know the following aspects.

The slope of the line can be calculated by dividing the y-intercept by the x-intercept of the line.
The area of the triangle formed by the line with the two coordinate axes is equal to half of the product of the x and y-intercepts.
The x and y-intercept are helpful to know the coordinates of the points where the line cuts the axis.
The x and y-intercept are useful to form the equation of a line, using the intercept form of the equation of a line.
X and y-intercept help us know the quadrant through which the line passes.

Solved Example of X and Y Intercept

Example 1: 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.

Example 2: Find the equation of the line, which makes intercepts –3 and 2 on the x- and y-axes respectively.

Solution: Given, a = –3 and b = 2.

By intercept form, we know that;

x/a + y/b = 1

x/-3 + y/2 = 1

2x – 3y + 6 = 0.

Hence, this is the required equation.

Example 3: A line passes through P (1, 2) such that its intercept between the axes is bisected at P. What is the equation of the line?

Solution: The equation of a line making intercepts a and b with x-axis and y-axis, respectively, is given by:

x/a + y/b = 1

1 = (a+0)/2 ⇒ a = 2

2 = (0 + b)/2 ⇒ b = 4

Therefore, the required equation of line is;

x/2 + y/4 = 1

⇒ 2x + y – 4 = 0

The y-intercepts are the points (0, 3/2) and (0, -3/2).

Practice Problems

Find the x-intercept and y-intercept for the line 5x – 8y = 2.
If the y-intercept of a line is -4 and the slope is 2/3, then write its equation.
What is the equation of a line whose x and y-intercepts are given as 1/3 and -3?

Example 1: Find x and y intercepts for the line 6x + 3y = 18.Solution:To find the x-intercept substitute y = 0.6x + 3(0) = 186x = 18x = 18/6 = 3The x-intercept is 3 and the line cuts the x-axis at the point (3, 0).Now to find the y-intercept put x = 0.6(0) + 3y = 183y = 18y = 18/3y = 6The y-intercept is 6, and the line cuts the y-axis at (6, 0).

Example 2: Draw the graph of 5x + 3y = 15 by finding the x and y-intercepts.Solution:To find the x intercept substitute y = 0.5x + 3(0) = 155x = 15x = 15/5 = 3The x-intercept is 3, and the line cuts the x-axis at (3, 0).To find the y intercept substitute x = 0 in the given equation.5(0) + 3y = 153y = 15y = 15/3 = 5The y-intercept is 5, and the line cuts the y-axis at (0, 5).

Frequently Asked Questions – FAQs

What is an intercept in Maths?

An intercept is a point where the straight line or a curve intersects the y-axis in a plane. It is also said to be a y-intercept.

What is the formula for y-intercept?

The formula for y-intercept is given by:
b = y – mx
Where b is the y-intercept and m is the slope of the line

What is the equation of the line with respect to x and y-intercepts?

The equation of the line, when a line is intersecting the x-axis and y-axis at point a and b respectively is given by:
x/a + y/b = 1

If a line intersects the y-axis at a point, then what is the value of x-coordinate?

If a line intersects the y-axis at a point, the value of the x-coordinate will be equal to zero.

What is the slope formula?

m=(y₂-y₁)/(x₂-x₁)

How do you find the x and y-intercept?

You can always find the X-intercept by setting Y to 0 in the equation and solve for X. Similarly, you can always find the Y-intercept by setting X to 0 in the equation and solve for Y.

What Are X And Y Intercept?

X and Y-intercept is the distance on the axis from the origin where a line or a curve cuts the coordinate axis of the graph. When a line or curve cuts the x-axis at a point, the distance of this point from the origin is called x-intercept and when that line or curve cuts the y-axis at a point, the distance of this point from the origin is called the y-intercept.

X and y intercepts can be found from the points where the line or curve cuts the coordinate axis. If the line cuts the x-axis at the point (a, 0), then ‘a’ is the x-intercept of the line, and if the line cuts the y-axis at the point (0,b), then ‘b’ is the y-intercept of the line.

X intercept is the x value of the point where the line cuts the x-axis, and the Y-intercept is the y value of the point where the line intersects the y axis. The x and y-intercept is also called the horizontal intercept and vertical intercepts, since it is situated on the horizontal axis, and vertical axis respectively.

What is the formula for X intercept form?

How do you find the X Intercept in y = mx + b? Here, y = mx + b is the slope-intercept form that can be used to identify the x and y-intercepts. The x-intercept is found by setting y = 0. On substituting the value of y =0, it gives the x-intercept value at which the line crosses the x-axis.

What is X and Y in slope-intercept?

The slope-intercept form of a linear equation is where one side contains just “y”. So, it will look like: y = mx + b where “m” and “b” are numbers. This form of the equation is very useful. The coefficient of “x” (the “m” value) is the slope of the line. And, the constant (the “b” value) is the y-intercept at (0, b)

How do you find X and Y with slope and y-intercept?

In an equation in slope-intercept form (y=mx+b) the slope is m and the y-intercept is b. We can also rewrite certain equations to look more like slope-intercept form. For example, y=x can be rewritten as y=1x+0, so its slope is 1 and its y-intercept is 0.

How do you find y-intercept in an equation?

Since the y-intercept always has a corresponding x-value of 0, replace x with 0 in the equation and solve for y. On a graph, the y-intercept can be found by finding the value of y when x=0. This is the point at which the graph crosses through the y-axis.

How to calculate x?

To solve for x, bring the variable to one side, and bring all the remaining values to the other side by applying arithmetic operations on both sides of the equation. Simplify the values to find the result. How do you get x by itself? Now, check the answer, x = 5 by substituting it back into the equation.

How do you solve for x?

One method is by using inverse operations. This means that you can use addition and subtraction to solve for x if the equation includes multiplication or division. For example, if you have the equation 4x=12, you can divide both sides by 4 to get x=3. Another method for solving for x is by using factoring.

What Is X And Y Intercept In Coordinate Geometry

The x and y intercept is the distance from the origin on the x-axis and y-axis where the line cuts the coordinate axis. If the line cuts the x-axis at the point (a, 0), then ‘a’ is the x-intercept of the line, and if the line cuts the y-axis at the point (0, b) then ‘b’ is the y-intercept of the line.

What Is The X And Y Intercept In An Equation

The x and y intercept in an equation of a line or a curve can be found by alternatively substituting x = 0 and y = 0 in the equation of the line. For a line having the equation ax + by + c = 0, the substitution of x = 0, and solving for y gives the y intercept as y = -c/b, and substitution of y = 0 and solving for x gives the x intercept as x = -c/a.

How to Find The X And Y Intercept For A Curve

The x and y-intercept for a curve can be found similar as we find for the line. The substitution of x = 0 in the equation of the curve and solving for the y value gives the y-intercept, and by substituting y = 0 in the equation of the curve and solving for the x value gives the x-intercept of the curve. The curve can have more than one x and y-intercepts.

How To Find Slope From X And Y Intercept?

The slope of the line is the negative of the y-intercept divided by the x-intercept. The slope of a line with x and y intercept is -y-intercept/x-intercept.

What Are The Instances Where There Are No X And Y Intercepts?

There are three instances when there is no x and y-intercept for the line. If the line is parallel to the x-axis, if the line is parallel to the y-axis, and if the line is passing through the origin. The line drawn parallel to the x-axis has only the y-intercept and does not have the x-intercept. And the line drawn parallel to the y axis has only the x-intercept and does not have the y intercept.

What Is The Use Of X And Y Intercept?

The x and y intercept is useful to find the slope of the line, to find the location of the line in the quadrants, and also to find the area of the triangle made by the line with the coordinate axis.

How to find the x and y intercept from an equation