If a particular point on a line and the slope of a line are known, we can write an equation for that line, using Point-Slope Form:

An equation of the line with slope m passing through the point is

Example. Write an equation of the line through (-5,6) with slope -2.

Solution. , and m = -2.

	Point-Slope Form
	Substitute values of , and m
y – 6 = -2(x + 5) y – 6 = -2x - 10	Simplify
y = -2x – 4	Solve for y

The equation of the line is y = -2x – 4.

The Point-Slope Form can also be used to find an equation for a line when two points on the line are known.

Example. Find an equation of the line through (1,-2) and (-3,-5).

Solution. We must first find the slope of the line, using the formula for slope:

	Slope formula
	Substitute:
	Simplify
m =
m =

Now we have the slope, m = , and a point, (1,-2). (We can use either of the given points, (1, -2) or (-3, -5), for the point.)

	Point-Slope Form
	Substitute:
	Simplify
	Solve for y

Another special form of a linear equation is the Slope-Intercept Form:

An equation of the line with slope m and y-intercept b is y = mx + b.

Example. Find an equation of the line with slope 4 and y-intercept -3.

Solution.

y = mx + b	Slope-Intercept Form
y = 4x – 3	Substitute: m = 4 and b = -3

Example. Find the slope and y-intercept of the line with equation 2x + 3y = 15.

Solution. We will start by solving the given equation for y to obtain Slope-Intercept Form.

2x + 3y = 15
3y = -2x + 15	Add –2x to both sides
	Divide both sides by 3
	Simplify

Comparing this result to Slope-Intercept Form, y = mx + b, we can determine that m = and b = 5.

Thus, the slope is and the y-intercept is 5.

Two other special forms of the linear equation are as follows:

Equation of a Vertical Line

An equation of the vertical line through the point (a, b) is x = a.

Equation of a Horizontal Line

An equation of the horizontal line through the point (a, b) is y = b.

Example. a) Find the equation of the vertical line through (3,-1).

b) Find the equation of the horizontal line through (-5, -3).

Solution. a) a = 3, so the equation is x = 3.

b) b = -3, so the equation is y = -3.

Slope and Parallel Lines

1. If two nonvertical lines are parallel, then they have the same slope.

2. If two distinct nonvertical lines have the same slope, then they are parallel.

3. Two vertical lines are parallel.

Example. Write an equation of the line passing through (-3,2) and parallel to the line with equation y = 2x + 1.

Solution. The equation y = 2x + 1 is in Slope-Intercept Form, with m = 2. So, the slope of this line is 2. Since our line is parallel to this given line, it must have the same slope (part 1 of Slope and Parallel Lines). Therefore, we are to find the equation of a line with slope 2 passing through the given point (-3,2).

	Point-Slope Form
	Substitute: m = 2,
y – 2 = 2(x + 3)	Simplify
y – 2 = 2x + 6
y = 2x + 8	Solve for y

The equation of the line is y = 2x + 8.

Slope and Perpendicular Lines

1. If two nonvertical lines are perpendicular, then the product of their slopes is –1.

2. If the product of the slopes of two lines is –1, then the lines are perpendicular.

3. A horizontal line is perpendicular to a vertical line.

Example. Write an equation of the line passing through (1,-3) and perpendicular to the line with equation -x + 2y = 1.

Solution. We can find the slope of the line -x + 2y = 1 using Slope-Intercept Form;

-x + 2y = 1	Solve for y
2y = x + 1	Add x to both sides
	Divide both sides by 2
	Simplify

From the Slope-Intercept Form, we determine that m = . The slope of the given line is . Because our line is perpendicular to this given line, the slope of our line is –2.

[(-2) = -1]. We now use Point-Slope Form with m = -2, , and .

	Point-Slope Form
	Substitute: m = -2, , and .
y + 3 = -2x + 2	Simplify
y = -2x – 1	Solve for y

The equation of the line is y = -2x – 1.