Equations of Lines

Calculating Slope

Any point on the xy-plane can be written as an ordered pair in the form (x, y). Taking two different points (x₁,y₁) and (x₂, y₂) on the line, the slope is:

\(m= \frac{\text{change in input}}{\text{change in output}}=\frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1}\)
where \(x_1\neq x_2\)

Example: Find the slope of the line through the points (2, 5) and (3,7).

Solution:

\(m = \frac{y_2-y_1}{x_2-x_1}\)
\(m = \frac{7-5}{3-2}\)
\(m=\frac{2}{1}\)
\(m=2\)

Equation of a line: Point-slope form

There are two common ways to represent a line. The point-slope form of the equation of a line is given by y – y₁ = m(x – x₁), where the line has a slope m and passes through the point (x₁, y₁).

Example: Write the equation of the line using point-slope form, given that the line passes through the point (-3, 8) and has a slope of 2. Also, graph the line.

Solution:

\(y-y_1 = m(x-x_1)\)
\(y-8=2(x+3)\)

Equation of a line: Slope-intercept form

Slope-intercept form of the equation of a line is given by y = mx + b, where the line has a slope m and a y-intercept b. The y-intercept is simply the place where the line crosses the y-axis. The corresponding ordered pair denoting the y-intercept is (0, y₁).

Example: Write the equation of the line with slope -3 and y-intercept 13 using the slope-intercept form.

Solution:

y = mx + b

y = -3x +13

Example: Find the equation of the line through the point (5, 3) and (0, 0).

Solution:

First, we find the slope:

\(m = \frac{y_2-y_1}{x_2-x_1}\)
\(m = \frac{3-0}{5-0}\)
\(m = \frac{3}{5}\)

Next, we find the equation of the line:

y - y₁ = m(x – x₁)

y – 0 = 3/5 (x – 0)

y = 3/5x