What is the X-Intercept?
The x-intercept is the location where the line crosses the x-axis of the plane. Thus, the value of the y-coordinate of a linear equation will always be equal to zero whenever it crosses the x-axis. For the x-intercept, the y-coordinate is zero, and for the y-intercept, the x-coordinate is zero. The x-intercept is also known as the horizontal intercept.
The equation of a line is given by:
y = mx + c
In general, the horizontal axis is commonly used to represent the variable x and the vertical axis to represent the variable y. Also, m is the slope of the line, and c is the y-intercept of the line.
The equation of a line is also given by:
Ax + By = C.
Here, A, B, and C are constants
Formula for the X-Intercept
The formula for the x-intercept of the line with the equation Ax + By = C, is,
x = \(\frac{C}{A}\)
The formula for the x-intercept for the line with the equation y = mx + c, is,
x = \(\frac{-C}{m}\)
Solved Examples
Example 1: Find the x-intercept of the equation, x + 3y = 12.
Solution:
To find the x-intercept, set y = 0 and solve for x.
x + 3 (0) = 12 [substitute zero for y]
x = 12 [Simplify]
Alternatively,
Comparing the given equation with the equation, Ax + By = C,
A = 1, B = 3 and C = 12
x-intercept = \(\frac{C}{A}~=~\frac{12}{1}~=~12\)
Hence, the x-intercept of the given equation is 12.
Example 2: Find the x-intercept of the equation, 2x + 4y = 8.
Solution:
Comparing the given equation with Ax + By = C,
A = 2, B = 4 and C = 8
x-intercept = \(\frac{C}{A}~=~\frac{8}{2}~=~4\).
Hence, the x-intercept of the given equation is 4.
Example 3: Find the x-intercept of the equation, y = 2x + 6.
Solution:
Comparing the given equation with y = mx + c ,
m = 2 and c = 6
x-intercept = \(\frac{-c}{m}~=~\frac{-6}{2}~=~-3\).
Hence, the x-intercept of the given equation is – 3.
Example 4: Rachel is trying to get the equation of a line with a slope of 6 and an x-intercept of -2. Can you assist her?
Solution:
A line with a slope m has the general equation y = mx + c.
The x-intercept of the line is – 2.
Applying the formula of the x-intercept,
x = – \(\frac{c}{m}\)
-2 = – \(\frac{c}{6}\) [Substitute the values of x and m]
c = 12 [Simplify]
If we combine the values of c and m, we get:
y = 6x +12
Hence, the equation of the line is y = 6x + 12.
When an equation is given in the form, y = mx + b, we can quickly determine the value of x by setting the value of y to 0. The obtained value of x is the x-intercept of the equation.
The x-intercept is the value of the x coordinate of a point on a plane where the value of the y coordinate is zero. It is the value of the x coordinate of the point where the graph intersects the x-axis.
Yes, 0 can be the x-intercept for the line y = mx, where ‘m’ denotes the slope of the line and x-intercept equals 0.