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# X and Y Intercept

X and Y-intercepts are the points on the coordinate axis where the graph of any given linear equation intersects. Intercepts are key concepts in the understanding of straight lines, which can reveal valuable information about the linear relationship of both variables. In this article, we will explore the definition of intercepts including both x and y intercepts, intercepts in graphs, and intercept formulas for various forms of line. By the end of this article, you’ll have a solid understanding of X-intercepts and Y-intercepts and how to use them in real-world applications.

## What are X and Y-Intercepts?

Intercepts are the points on the x and y-axis respectively where any linear equation intersect. The following diagram shows a line with x and y-intercepts.

### X-Intercept

The point of intersection of a line and the x-axis is called the x-intercept and in the above diagram, Point A(a, 0) represents the x-intercepts of the line. The y-coordinate of the x-intercept is always 0. Thus, the general form of x-intercept is (x, 0)

### Y-intercept

The point of intersection of a line and the y-axis is called the y-intercept and in the above diagram, Point B(0, b) represents the y-intercepts of the line. The x-coordinate of the y-intercept is always 0. Thus, the general form of y-intercept is (0, y)

## How to Find X-Intercepts and Y-Intercepts?

For any general equation of a straight line equation i.e., Ax + By = C.

Divide the equation by C,

(Ax/C) + (By/C) = C/C

Now, putting y = 0, for the x-intercept we get

Thus, the x-intercept is (C/A, 0).

Now, putting x = 0, for the y-intercept we get

Thus, the y-intercept is (0, C/B)

## Intercepts From of Line

Intercept From of Line is,

x/a + y/b = 1

Where,

• a is the x-intercept of the line
• b is the y-intercept of the line

### Finding Intercept Form Slope-Intercept Form of Line

The slope-Intercept Form of a straight line is,

y = mx + c

Where,

• (0, c) is the y-intercept,
• m is the slope of the given line.

For, this form of the line

• x-intercept is given by  (-c/m, 0)
• y-intercept is given by (0, c)

### Finding Intercept Form Point-Slope Form of Lines

The point-slope form of a line is given as follows:

y – y1 = m(x – x1)

where:

• (x1, y1) is a point on the line
• m is the slope of the line.

To find, the x and y-intercepts of the given line,

Here, rearranging the equation, we get

y = mx – mx1 + y1

â‡’ y = mx + (-mx1 + y1)

Comparing it with y = mx + c, we get

c = -mx1 + y1, which is the y-intercept of the given line.

and x-intercept is -c/m = (mx1 – y1)/m = x1 – y1/m

Thus, x and y-intercept of the given y – y1 = m(x – x1) are

• x-Intercept is given by  x1 – y1/m
• y-Intercept is given by  -mx1 + y1

### Finding Intercept Form Two Point Form of Line

If in the Point-slope Form of a line, we substitute the formula for slope, m = (y2 – y1)/(x2 – x1) we get the two-point Form of a Line, i.e.,

y – y1 = (y2 – y1)/(x2 – x1)(x – x1)

where,

• (x1, y1) and (x2, y2) are the two points from which line passes through.

Thus, x and y-intercept of the given y – y1 = (y2 – y1)/(x2 – x1)(x – x1) are

• y-Intercept is given by x1 – y1(x2 – x1)/(y2 – y1)
• y-Intercept is given by -x1(y2 – y1)/(x2 – x1) + y1

## Uses of X and Y Intercept

X and Y Intercepts have various uses and some of them are,

• In curve tracing, for example, we have an unknown curve, so intercepts are one of the first parameters in the analysis of the curve.
• Both intercepts in whichever quadrant forms a triangle, whose area can calculate by 1/2 times the product of intercepts.
• By plotting both intercepts on the coordinated axes, we can plot the graph of the linear equation.

## Sample Problems on X and Y Intercept

Problem 1: Find the x and y-intercepts of the line having equation: y = x + 10

Solution:

Converting the equation of the given line in intercept form:

y – x = 10

â‡’ (y/10) – (x/10) = 1, ———dividing both sides by 10

â‡’ (y/10) + (-x/10) = 1

â‡’ (x/(-10)) + (y/10) = 1,

Thus, x-intercept is -10 and y intercept is 10.

Another solution:

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

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

0 = x + 10

â‡’ x = -10

Thus, x-intercept of given line is -10.

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

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

y = 0 + 10

â‡’ y = 10

Thus, y-intercept of given line is 10.

Problem 2: Find the x and y-intercepts of the line having equation: 20y = 10 – 40x

Solution:

Converting the equation of the given line in intercept form :

20y + 40x = 10

â‡’ (20y/10) + (40x/10) = 1, ———dividing both sides by 10

â‡’ (2y/1) + (4x/1) = 1

â‡’ (x/(1/4)) + (y/(1/2)) = 1,

Thus, x-intercept is (1/4) and y intercept is (1/2).

Another solution:

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

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

20Ã—(0) = 10 – 40x,

â‡’ 0 + 40x = 10,

â‡’ x = 1/4

Thus, x-intercept of given line is 1/4 or 0.25

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

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

20y = 10 – 40Ã—(0)

â‡’ 20y = 10,

â‡’ y = 1/2

Thus, y-intercept of given line is 1/2 or 0.5

Problem 3: Find the x and y-intercepts of the line having equation: 4x + 5y = -3

Solution:

Converting the equation of the given line in intercept form :

4x + 5y = -3 ——-given

â‡’ 4x/(-3) + 5y/(-3) = -3/(-3), —–dividing both sides by -3

â‡’ x/(-3/4) + y/(-3/5) = 1,

Thus, x-intercept is (-3/4) and y intercept is (-3/5)

Another solution:

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

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

4x + 5Ã—(0) = -3,

â‡’ 4x + 0 = -3,

â‡’ x = -3/4

Thus, x-intercept of given line is -3/4

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

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

4Ã—(0) + 5y = -3,

â‡’ 0 + 5y = -3,

â‡’ y = -3/5

Thus, y-intercept of given line is -3/5

Problem 4: A line AB has x-intercept = 0. Find its y-intercept.

Solution:

x-intercept of given line is 0.

This means that the point of intersection of the given line and X-axis is (0, 0).

In other words, the given line passes through the origin.

Thus, y-intercept of the given line is 0 (as the point of intersection of the given line and Y-axis is also (0, 0)).

Problem 5: A line passes through the point (3, 4), (p, q), and (c, d), where p and d are x and y-intercepts respectively. Find the value of p, q, c, and d given that the slope of the line is -1/2.

Solution:

p is the x-intercept of the given line, so (p, q) lies on X-axis.

This means that q = 0 ——–(i)

d is the y-intercept of the given line, so (c, d) lies on Y-axis.

This means that c = 0 ——–(ii)

Slope of any line = (y2 – y1)/(x2 – x1), where (x1, y1) and (x2, y2) are two points that lie on it.

Slope of given line = (4-q)/(3-p), or

-1/2 = (4-0)/(3-p), ——from (i)

â‡’ (-1)Ã—(3-p) = 4Ã—2,

p – 3 = 8, or

Thus, p = 11  ——–(iii)

Slope of given line = (4-d)/(3-c), or

-1/2 = (4-d)/(3-0), ——-from (ii)

â‡’ (-1/2)Ã—3 = 4 – d,

â‡’ d – 3/2 = 4,

â‡’ d = 4 + 3/2

Thus d = 11/2 or 5.5 ——–(iv)

Thus, the values are : p = 11, q = 0, c = 0, d = 11/2

## FAQs on X and Y-intercepts of a Straight Line

### Q1: What is a Straight Line?

• “The shortest distance between any two points is known as a line.”
•  “A widthless length is called a line.”

### Q2: What is an X-Intercept of a Straight Line?

The point of intersection between line and x-axis is called x-intercept.

### Q3: What is a Y-Intercept of a Straight Line?

The point of intersection between line and y-axis is called y-intercept.

### Q4: How to Find the X-intercept of a Straight Line?

As x-intercept is the point where the line intersects the x-axis. Thus, we can find the coordinate of the x-intercept by putting (x,0) in the given linear equation and then finding the value of x. Thus, (x, 0) is the coordinate of the x-intercept.

### Q5: How to Find the Y-intercept of a Straight Line?

As y-intercept is the point where the line intersects the y-axis. Thus, we can find the coordinate of the y-intercept by putting (0, y) in the given linear equation and then finding the value of y. Thus, (0, y) is the coordinate of the y-intercept.

### Q6: Can a Straight Line have both X and Y-intercepts?

Yes, all straight lines, except horizontal and verticles lines, have both x and y-intercepts.

### Q7:  Can a Straight Line have only an X-intercept?

Yes, a straight line can have only an x-intercept. In the case of the verticle line (parallel to the y-axis), it only intersects the x-axis as it is parallel to the y-axis.