5.3 Writing Linear Equations given Two points

Objective Write an equation of a line given two points of the line.

Steps:

1. Find the slope.

m = y - y

12

x - x2 1

2. Find the y-intercept. Substitute the slope m and the coordinates of one of the points into the slope- intercept form, y = mx + b, and solve for the y-intercept b.

3. Write an equation of the line. Substitute the slope m and the y-intercept b into the slope-intercept form, y = mx + b.

EXAMPLE

Write an equation of the line that passes through the points(1, -3) and (3, 5).

1. Find the slope. m = 5 – (-3)

3 – 1m = 8

2m = 4

2. Find the y-intercept. Choose one of the ordered pairs.(Either ordered pair will result in the same y-intercept)

Using (1, -3) y = mx + b

-3 = 4(1) + b-3 = 4 + b -4 -4-7 = b

Using (3, 5)y = mx + b

5 = 4(3) + b5 = 12 + b

-12 -12-7 = b

(x1,y1) (x2,y2)

3. Substitute the slope and the y-intercept into y = mx + b

m = 4 and b = -7

y = 4x - 7

EXAMPLE Your Turn

1. Find the equation of a line passing through the points (1, 6) and (3, -4).

Steps1) Find slope

2) Find y-intercept b

3) Write equation

m = -5b = 11

Y = -5x +11

Two different nonvertical lines are perpendicular if and only if their slopes are opposite reciprocals of each other.

Negative Reciprocal Examples

Number Negative Reciprocal

3 3

1

2

1-2

4

3

3

4

EXAMPLE

Write the equation of a line perpendicular to the liney = -2x + 2 and passing through (6, 8).

m = -2. Perpendicular slope = ½.

Replace ½ into y =mx +b then find b Use (6,8) for (x,y)

8 = ½(6) + b8 = 3 + b

-3 -35 = b

The equation of the perpendicular line is y = ½ x + 5.

EXAMPLE Try this.

Find the equation of a line perpendicular to the equationy = -1/3x – 2 and through the point (-2, -3).

m = 3

y = mx + b

-3 = 3(-2) + b

-3 = -6 + b 6 6

3 = b

y = 3x + 3

Objective 22 Modeling a real-life situation.

A camp program charges a registration fee and a daily amount.If the total bill for one camper was $338 for 12 days and the totalbill for another camper was $506 for 19 days, how much will the bill be for a camper who enrolls for 30 days?

Identify the independent variable (x) and the dependent variable (y).Since part of the bill depends on the number of days, the independent variable is the number of days.

Write the two ordered pairs. (12, 338) and (19, 506)

The registration fee is the y-intercept (b). This value is not know.

Find the slope and the y-intercept.(12, 338) (19, 506)

506 - 338m = 19 - 12 m = 24

y = mx + b

338 = 24(12) + b338 = 288 + b-288 -288

50 = b

y = 24x + 50 This means that the camper pays $24 each day and aregistration fee of $50.

If a camper is there for 30 days, find the cost.y = 24(30) + 50y = 720 + 50y = $770