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

Therefore, the equation of the line is y = 3/5x.

Equations of a Line: Given Two Points

Equations of a Line: Parallel Lines

Equation of a Line: Parallel Lines #2

Equations of a Line: Perpendicular Lines

Equation of a Line: Perpendicular Lines #2