Finding Equations of Lines
 How to find the equation of a line from a graph.
Method 1:
1) Look at the graph. Locate 2 coordinates on the line and find the slope by using the
y  y1
rise
ratio of
or the slope formula of: m= 2
.
x2  x1
run
2) Find the coordinate where the line crosses the y-axis: (0,b)
3) Substitute your slope “m” and y-intercept “b” into the slope-intercept formula:
y=mx+b.
Example:
2 coordinates: (0,0) and (3,4). Slope=
4
3
y-intercept=(0,0)
4
4
The equation of the line is: y= x+0 or y= x
3
3
Method 2:
1) Look at the graph. Locate 2 coordinates on the line and find the slope by using the
y  y1
rise
ratio of
or the slope formula of: m= 2
.
x2  x1
run
2) Use the point slope formula y-y1=m(x-x1)
Substitute slope value for M and pick either point to get the values for y1 and x1.
Solve the equation for y.
Example: using the graph above
4
4
The slope is
and use the coordinate (0,0): y-0= (x-0)
3
3
4
4
The equation of the line is y= x+0 or y= x.
3
3
 How to find the equation of a line from a table.
1) Look at the table. Locate 2 coordinates in the table and find the slope by using the
y  y1
slope formula of: m= 2
.
x2  x1
2) Use the point slope formula y-y1=m(x-x1)
Substitute slope value for M and pick either point to get the values for y1 and x1.
3) Solve the equation for y.
Example:
86
1) 2 coordinates: (3,6) and (4,8): m=
=2, m=2
x
y
43
2) Substitute (3,6) and m=2 into y-y1=m(x-x1):
3
6
y-6=2(x-3)
4
8
y-6=2x-6
5
10
+6
+6
3) The equation of the line is y=2x+0 or y=2x
 How to find the equation of a line (closed form) from arithmetic sequences
1) Take the arithmetic sequence and place into a t-chart. The x values are the term
positions (1, 2, 3, etc…), the y values are the sequence numbers in that term
position.
2) Look at the table. Locate 2 coordinates in the table and find the slope by using the
y  y1
slope formula of: m= 2
.
x2  x1
3) Use the point slope formula y-y1=m(x-x1)
Substitute slope value for M and pick either point to get the values for y1 and x1.
4) Solve the equation for y.
Example:
Sequence: -4, 1, 6, 11, 16, …
x
y
1
-4
2
1
3
6
4
11
5
16
6 1
=5
32
2) Substitute (3,6) and m=5 into y-y1=m(x-x1):
1) (2,1) and (3,6) : m=
3) Solve for y:
y-6=5(x-3)
y-6=5x-15
y=5x-9
 How to find the equation of a line in context
Step 1: Identify the rate of change (slope or m) and the starting value (y-intercept or b).
Step 2: Write a verbal model. Then write an equation.
Step 3: Calculate the rate of change, also known as slope, also known as m
Step 4: Evaluate using the point-slope formula.
Example 1:
Your gym membership costs $35 per month after an initial membership fee. A member
has paid a total of $250 after 6 months. Write an equation that gives the total cost of a
gym membership as a function of the length of membership (in months). Find the total
cost of membership after 10 months.
1) Rate of change: monthly cost= $35 per month
Starting value: initial membership fee … the point is (0,b)
2) Total Cost (C)=Monthly cost ($35) X Number of months (t)+Membership Fee (b)
C=35t+b
3) Substitute: 250=35(6)+b; b=40
4) Substitute to evaluate: C=35(t)+40;when t=10, C=390
Example 2:
The initial fee to have a website set up using a server is $48. It costs $44 per month to
maintain the website.
1) Rate of change: $44
Starting value: set up=$48… so our point is (0,48)
2) Total Cost (C)=Monthly cost ($44) x Number of months (t)+Set up fee($48)
C=44t+48
 How to find the best fit line (equation of a line) from a scatterplot
Step1: Make a scatter plot of data or use an existing one.
Step 2: Decide whether the data can be modeled by a line (has a positive or negative correlation)
Step 3: Draw a line that appears to fit the data closely. There should be approximately as many points
above the line as below it.
Step 4: Write an equation by choosing 2 coordinates on the line. Find the slope and the y-intercept:
A. Look at the graph. Locate 2 coordinates on the line and find the slope by using the ratio of
the slope formula of: m=
y2  y1
.
x2  x1
B. Substitute 1 coordinate and the slope into the point-slope form:
rise
or
run
y-y1=m(x-x1):, then solve for y.
Example:
A. 2 coordinates: (1, 1.5) and (2.5, 2.5)=
2.5  1.5
1
10 2
=
or

2.5  1 1.5 15 3
2
B. y-y1=m(x-x1):; y-1.5= (x-1)
3
2
5
C. y= x+
3
6
Homework: Create one of each.
Directions: Create one of each type of method for finding an equation of a line.
Show an example, the method for finding the equation, and give the equation. You
will need graph paper, knowledge of graphs, scatterplot, sequences, and tables, and a
lot of creativity for this assignment.