Intercept Form of Linear Equations

Intercept From of Line is,

x/a + y/b = 1

Where,

• a is the x-intercept of the line which means if we put x = 0, in the above line, we get the solution for y as b
• b is the y-intercept of the line means if we put y = 0, we get the solution as x = b

Some examples of the linear equation in intercept form for two variables are,

• x/2 + y/3 = 1
• x/3 - y/2 = 1

What are x and y intercepts?

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

X-Intercept : Point A(a, 0) represents the x-intercepts of the line where the y-coordinate is always 0. Thus, the general form of x-intercept is (x, 0)

Y-intercept : Point B(0, b) represents the y-intercepts of the line where x coordinate is 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

\Rightarrow \frac{x}{C/A} + \frac{y}{C/B} = 1

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

 \Rightarrow \frac{x}{C/A} + \frac{0}{C/B} = 1 \\ \Rightarrow x = C/A

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

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

\Rightarrow \frac{0}{C/A} + \frac{y}{C/B} = 1 \\ \Rightarrow y = C/B

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

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

⇒ 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.

Read More,

• Equation of a Straight Line
• Point Slope Form
• Linear Equations

Examples on X and Y Intercept

Example 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

⇒ (y/10) - (x/10) = 1, ---------dividing both sides by 10

⇒ (y/10) + (-x/10) = 1

⇒ (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

⇒ 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 

⇒ y = 10

Thus, y-intercept of given line is 10.

Example 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

⇒ (20y/10) + (40x/10) = 1, ---------dividing both sides by 10

⇒ (2y/1) + (4x/1) = 1

⇒ (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:

20×(0) = 10 - 40x,

⇒ 0 + 40x = 10, 

⇒ 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 - 40×(0)

⇒ 20y = 10,

⇒ y = 1/2

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

Example 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

⇒ 4x/(-3) + 5y/(-3) = -3/(-3), -----dividing both sides by -3

⇒ 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 + 5×(0) = -3,

⇒ 4x + 0 = -3, 

⇒ 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:

4×(0) + 5y = -3, 

⇒ 0 + 5y = -3,

⇒ y = -3/5

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

Example 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)).

Example 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)

⇒ (-1)×(3-p) = 4×2,

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)

⇒ (-1/2)×3 = 4 - d,

⇒ d - 3/2 = 4,

⇒ 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

Practice Questions on X and Y Intercept

Question 1: Find the x-intercept and y-intercept of the equation 2x + 3y = 6.

Question 2: Determine the x-intercept and y-intercept of the equation 4x - y = 8.

Question 3: What are the x-intercept and y-intercept of the equation 5x + 2y = 10?

Question 4: Find the x-intercept and y-intercept of the equation y = -3x + 7.

Question 5: Determine the x-intercept and y-intercept of the equation x+y = 5.