X and Y Intercept Formula

X and Y Intercept Formula as the name suggests, is the formula to calculate the intercept of a given straight line. An intercept is defined as the point at which the line or curve intersects the graph's axis. The intercept of a line is the point at which it intersects the x-axis or the y-axis.

When an equation isn't in the form y = mx + b, we determine the intercepts by substituting 0 as necessary and solving for the corresponding variable.

• To find the y-intercept, we set x = 0 and solve for y. This yields the point (0,y).
• To find the x-intercept, we set y = 0 and solve for x. This gives us the point (x,0).

Intercept Definition in X and Y Intercept

An intercept is defined as the point where the line or curve crosses the axis. If the point is on the x-axis then it is called the x-intercept and if the point is on the y-axis, then it is called the y-intercept.

We generally represent the x-intercept by a and the y-intercept by b. The equation of the line making a and b intercept on the x and y axis respectively is,

x/a + y/b = 1

What is X-Intercept?

The x-intercept of a line is the point at which the line intersects the x-axis. So, to find the x-intercept put y = 0 in the equation of a line. 

X-Intercept Formula

The formula of x-intercept for slope-intercept equation y = mx + c is given by,

👁 X-Intercept-Formula

x-intercept = -c/m

Thus, (-c/m, 0) is the coordinate of x-intercept.

Where,

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

Derivation of X-Intercept Formula

Consider a line given in the slope intercept form y = mx + c, where the line has a intercept (c, 0) and has a slope m.

Put y = 0 in the equation to get the x-intercept.

⇒ 0 = mx + c

Solve the equation for x.

⇒  mx = -c

⇒ x = -c/m

This derives the formula for x-intercept.

Check: Slope Intercept Form of line

What is Y-Intercept?

The y-intercept of a line is the point at which the line intersects the y-axis. So, to find the y-intercept, put x = 0 in the equation of a line. 

Y-Intercept Formula

The formula of y-intercept for slope-intercept equation y = mx + c is given by,

👁 Y-Intercept-Formula

y-intercept = c

Thus, (0, c) is the coordinate of y-intercept.

Derivation of Y-Intercept Formula

Consider a line given in the slope-intercept form y = mx + c, where the line passes through the point (0, c) and has a slope m.

Put x = 0 in the equation to get the y-intercept.

⇒  y = m (0) + c

⇒  y = 0 + c

⇒  y = c

This derives the formula for y-intercept.

How To Find X And Y Intercepts?

To find the x-intercept we put y = 0 in the given function and then solve for x. The resultant value of x is the x-intercept of the given function.

Example: Find the x-intercept of the linear equation 2x + 3y = 7.

Solution:

For the x-intercept of the linear equation 2x + 3y = 7

Put y = 0,

2x + 3×0 = 7

⇒ x = 7/2

Thus, the x-intercept of 2x + 3y = 7 is 7/2.

To find the y-intercept we put x = 0 in the given function and then solve for y. The resultant value of y is the y-intercept of the given function.

Example: Find the y-intercept of the linear equation 3x + 4y = 12.

Solution:

For the y-intercept of the linear equation  3x + 4y = 12

Put x = 0,

3×0 + 4y = 12

⇒ y = 12/4

⇒ y = 3

Thus, the y-intercept of 3x + 4y = 12 is 3.

Intercept Form of a Straight Line

Intercept Form of a Straight Line, mathematically given by 

x/a + y/b = 1

Where, 

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

Intercept Graph

We know that the intercept is the points on the axes that are cut by a straight line. The point on the x-axis is called the x-intercept, and the point on the y-axis is called the y-intercept. The image added below shows the line, with x and y intercepts.

👁 X and Y Intercept Formula

For Point-Slope Form

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 x₁ - y₁/m and  -mx₁ + y₁ respectively.

Uses of X And Y Intercept

There are various use cases of X And Y Intercepts, some of which are as follows:

• We use the concept of intercepts 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.

Articles related to X and Y Intercept Formula:

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

Solved Example of X and Y Intercept Formula

Problem 1: Calculate the x-intercept of the equation x + 3y = 8.

Solution:

We have the equation as, x + 3y = 8.

Put y = 0 to find the x-intercept and then solve the equation for x.

⇒  x + 3 (0) = 8

⇒  x = 8

So, the x-intercept for the equation is (8, 0).

Problem 2: Calculate the x-intercept of equation 4x + 7y = 10.

Solution:

We have the equation as, 4x + 7y = 10.

Put y = 0 to find the x-intercept and then solve the equation for x.

⇒  4x + 7 (0) = 10

⇒  4x = 10

⇒ x = 10/4

⇒  x = 5/2

So, the x-intercept for the equation is (5/2, 0).

Problem 3: Calculate the y-intercept of equation 4x + 3y = 24.

Solution:

We have the equation as, 4x + 3y = 24.

Put x = 0 to find the y-intercept and then solve the equation for y.

⇒ 4(0) + 3y = 24

⇒  3y = 24

⇒  y = 24/3

⇒  y = 8

So, the y-intercept for the equation is (0, 8).

Problem 4: Calculate the y-intercept of equation 8x + 5y = 25.

Solution:

We have the equation as, 8x + 5y = 25.

Put x = 0 to find the y-intercept and then solve the equation for y.

⇒  8(0) + 5y = 25

⇒ 5y = 25

⇒  y = 25/5

⇒  y = 5

So, the y-intercept for the equation is (0, 5).

Problem 5: Calculate the x- and y-intercept of equation 4x² + 9y² = 25.

Solution:

We have the equation as, 4x² + 9y² = 25.

Put y = 0 to find the x-intercept and then solve the equation for x.

⇒  4x² + 9 (0)² = 25

⇒  4x² = 25

⇒  x² = 25/4

⇒  x = ±5/2

So, the x-intercept for the equation is (±5/2, 0).

Put x = 0 to find the y-intercept and then solve the equation for y.

⇒  4 (0)² + 9y² = 25

⇒  9y² = 25

⇒  y² = 25/9

⇒ y = ±5/3

So, the y-intercept for the equation is (0, ±5/3).

Practice Problems on X and Y Intercept Formula

1. Find the x and y intercepts of the equation 3x - 2y = 6

2. Determine the x and y intercepts of the line represented by the equation 2y + 4x = 8

3. Find the x and y intercepts of the equation y = 2x - 3

4. Determine the x and y intercepts of the line represented by the equation 4x + 3y = 12

5. Find the x and y intercepts of the equation 5y - 2x = 10