The Y-intercept Formula
The y-intercept of a line is the value of the y-coordinate corresponding to which the value of the x-coordinate will be zero. We can determine the y-intercept of a line from its graph by observing the point at which the graph of the line intersects the y-axis. The value of the y-coordinate at this point of intersection is the y-intercept of the line.
We can determine the y-intercept of a line from its equation as well. The formula for the y-intercept of a straight line depends upon the form of the equation of the line.
If the equation of the line is in the standard form, that is, Ax + By = C, then the formula for the y-intercept is:
y = \(\frac{C}{B}\) Where,
- y is the y-intercept
- B and C are the constants
If the equation of the line is in the slope-intercept form, that is,
y = mx + c, then, the y-intercept formula is:
y = c
Where,
- m = slope of the line
- c = y-intercept of the line
If the equation of the line is in the point-slope form, that is, y – b = m(x – a) then, the y-intercept formula is:
y = -am + b
Where,
- m is the slope of the line
- (a, b) are points on the line
Solved Examples
Example 1: Find the y-intercept of the equation 5x + 2y = 12.
Answer: The given equation 5x + 2y = 12 is in the standard form.
So the y intercept is:
y = \(\frac{C}{B}\) Formula for y-intercept
y = \(\frac{12}{2}\) Substitute 12 for C and 2 for B
y = 6 Solve
Hence, the y-intercept is 6.
Example 2: Find the equation of the line in slope-intercept form with a slope of -3 and a y-intercept of 4.
Answer: As we have learned, a linear equation in the slope-intercept form is, y = mx + b
Here, the slope, m = -3 and y-intercept, b = 4
So, substituting these values in the slope-intercept equation form and by simplifying,
y = mx + b
= (-3)x + 4
y = -3x + 4
Hence, the equation of the line in slope-intercept form is y = -3x + 4
Example 3: Engineers are constructing a road that is 350 miles long. After 2 weeks, 150 miles of road still need to be constructed. How much time in total will it take to complete the road?
Answer: Let us write the equation of a line representing road length y (in miles) remaining after x weeks.
At the start of the construction, 350 miles needed to be constructed, that is, (0, 350). So the y-intercept is 350.
After 2 weeks, 150 miles still need to be constructed, that is, (2, 150).
Here, we have two points (0, 350) and (2, 150).
So the slope, m = \(\frac{\text{change in y}}{\text{change in x}}\)
m = \(\frac{-200}{2}=-100\)
So the equation of the line is y = -100x + 350
When the road is completed y = 0, that is,
0 = -100x + 350
-350 = -100x [Subtract 350 from both sides]
3.5 = x [Divide each side by -100]
Therefore it takes 3.5 weeks to complete the construction of the road.
The y-intercept of an equation represents the point where the graph of the equation intersects the y-axis.
The graph of a linear equation is a straight line.
The slope of a line is the ‘ratio of rise to run’ between any two points on the line.