Home / United States / Math Classes / Formulas / Y Intercept Formulas

The y-intercept of a linear equation is the point at which the graph of the equation crosses the y-axis. Here we will discuss the formula for the y-intercept as this calculation is essential to solve problems on lines and graphs....Read MoreRead Less

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

**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.

Frequently Asked Questions

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.