Finding the Equation of a Line: Given Two Points

Finding the Equation of a Line: Given Two Points

Goal: To learn how to write an equation for a line passing through two points.

To write the equation of a line passing through two points, follow these steps: x1y1, x2,y2
Step 1: Calculate the slope, m=y2-y1x2-x1
Step 2: Find the yintercept, b, by substituting the slope and one of the points into y=mx+b
Step 3: Write the equation by substituting the slope and the yintercept, b, into y=mx+b
Example 1: Determine the equation of the line passing through the points (1, 2) and (3, 10)
Solution:
1. Calculate the slope
(1,2) and (3,10)
m=y2-y1x2-x1
=10-23-1
=82
=4
2. Find the y-intercept, b
y=mx+b
y=4x+b...(*)
-Substitute one of the points into(*)
(1,2)
y=4x+b
2=4(1)+b
2=4+b
2-4=b
-2=b
b=-2
3. Substitute m=4
and b = - 2 in y=mx+b
∴y = 4x – 2
Equation of the line passing through (1,2) and (3,10).
OR if the ather point was used.
(3,10)
y = 4x + b
10 = 4(3) + b
10 = 12 + b
b + 12 = 10
b = 10 – 12
b = - 2
Example 2: Jake R worked for 5 hours and earned .

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Jake C worked for 8 hours and earned $84.If they both earn the same hourly wage, determine the equation that represents their total earnings, E, compared to the total hours, t, worked. Write the equation in the form E = mt + b.
Solution: (x,y), where se rep. hours worked and y amount of money earned
Jake R. (5, 60) (x11 ,y1 )
Jake C. (8, 84) x2,y2
1. Calculate the slope, m
m=y2-y1x2-x1
=84-608-5
=243
=8

3.Sub. m = 8 and b=20
in y = mx + b OR
E = mt + b
∴E = 8t + 20
2. Calculate the y-intercept, b
y = mx + b
y = 8x + b Sub, m = 8
Use one of the points,5,60
60=85+bSub.x=5 and y=60
60 = 40 + b Solve for b
60 - 40 = b
20 = b
b = 20
Example 3: Determine the equation of the line that has an xintercept of 3 and a yintercept of 8.
Solution
x-intercept is -3. As a point (-3,0)
y-intercept is 8. As a point (0, 8)
Write the equation of the line passing through the points (-3,0) and (0,8).

1. Calculate the slope
m=y2-y1x2-x1
=8-00--3
=80+3
=83
2. The y-intercept is 8.
So, b=8

3. Substitute m=83 and b=8 in y=mx+b
∴ y=83x+8
Example 4: The graph illustrates a walker’s movement in front of a motion sensor. Answer true or false to the following statements, and explain your answers.
a) The walker started at a distance of 1 m from the sensor
Fulse since the walker
Started 7 m away from the sensor.
b) After 3 s, the walker was 1 m from the sensor.
True since point (3,1) is on the path and it means that offer 3 seconds the walker was 1m from the sensor.
c) The walker's speed was 2 m/s toward the sensor.
(3,1) (1, 5)
m=5-11-3
=4-2
=-2
True Since m = - 2
-2=-21 This means that for every second, the walker gets closer to the Sensor by 2 meters.
The method you follow to write an equation for a line differs depending on the information you are given. Summarize the steps you would use to write the equation of a line given the following information:
a) two points on the line
1. Calculate the slope, m=y2-y1x2-x1
2. Calculate the y-intercept, b
3. Sub. m and b in y = mx + b
b) one point on the line and its y-intercept
1. Write the y-intercept as a point, (0,5).
2. Calculate the slope, m=y2-y1x2-x1
3. Sub. mand bin y = mx + b

c) the x- and y-intercepts
1. Write the intercepts (x,0) and (0,y)
2. Calculate the slope, m=y2-y1x2-x1
3. Calculate the y-intercept, b
4. Sub. m and b in y = mx + b
d) the slope and the y-intercept
-Substitute m and b in y=mx+b
e) the slope and a point on the line
1. Find the vertical intercept (y-intercept), b
2. Substitute m and b in y=mx+b