How to Determine the Equation of a Line

There are many instances in science and math in which you will need to determine the equation of a line. In chemistry, you'll use linear equations in gas calculations, when analyzing rates of reaction, and when performing Beer's Law calculations. Here are a quick overview and example of how to determine the equation of a line from (x,y) data.

There are different forms of the equation of a line, including the standard form, point-slope form, and slope-line intercept form. If you are asked to find the equation of a line and are not told which form to use, the point-slope or slope-intercept forms are both acceptable options.

Standard Form of the Equation of a Line

One of the most common ways to write the equation of a line is:

Ax + By = C

where A, B, and C are real numbers

Slope-Intercept Form of the Equation of a Line

A linear equation or equation of a line has the following form:

y = mx + b

m: slope of the line; m = Δx/Δy

b: y-intercept, which is where the line crosses the y-axis; b = yi - mxi

The y-intercept is written as the point (0,b).

Determine the Equation of a Line - Slope-Intercept Example

Determine the equation of a line using the following (x,y) data.

(-2,-2), (-1,1), (0,4), (1,7), (2,10), (3,13)

First calculate the slope m, which is the change in y divided by the change in x:

y = Δy/Δx

y = [13 - (-2)]/[3 - (-2)]

y = 15/5

y = 3

Next calculate the y-intercept:

b = yi - mxi

b = (-2) - 3*(-2)

b = -2 + 6

b = 4

The equation of the line is

y = mx + b

y = 3x + 4

Point-Slope Form of the Equation of a Line

In the point-slope form, the equation of a line has slope m and passes through the point (x₁, y₁). The equation is given using:

y - y₁ = m(x - x₁)

where m is the slope of the line and (x₁, y₁) is the given point

Determine the Equation of a Line - Point-Slope Example

Find the equation of a line passing through points (-3, 5) and (2, 8).

First determine the slope of the line. Use the formula:

m = (y₂ - y₁) / (x₂ - x₁)
m = (8 - 5) / (2 - (-3))
m = (8 - 5) / (2 + 3)
m = 3/5

Next use the point-slope formula. Do this by choosing one of the points, (x₁, y₁) and putting this point and the slope into the formula.

y - y₁ = m (x - x₁)
y - 5 = 3/5 (x - (-3))
y - 5 = 3/5 (x + 3)
y - 5 = (3/5)(x + 3)

Now you have the equation in point-slope form. You could proceed to write the equation in slope-intercept form if you wish to see the y-intercept.

y - 5 = (3/5)(x + 3)
y - 5 = (3/5)x + 9/5
y = (3/5)x + 9/5 + 5
y = (3/5)x + 9/5 + 25/5
y = (3/5)x +34/5

Find the y-intercept by setting x=0 in the equation of the line. The y-intercept is at the point (0, 34/5).