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Chapter 4
Analyzing Linear Equations
4.1 Rate of Change & Slope
• Rate of Change- a ratio that describes, on
average, how much one quantity changes
compared to another
Change in y
Rate of change=
x
Change in x
y
Negative slope
Ex:
Time
Walking(sec)
+1
+1
+1
Distance
Walked(ft)
1
4
2
8
3
12
4
Positive slope
16
4
+4
= 4 ft/sec
Rate of change =
+4
+4
1
Zero slope
Undefined slope
• Slope- the ratio of the change in y-coordinates
over the change in x-coordinates for a line
rise
Slope=
Slope:
y2 – y1
run
m=
x2 – x1
Ex: Find the slope (4, 6) (-1, 2)
x1
y1
x2
y2
Ex:
26 4 4
m


1  4  5 5
Ex:
m
5  5 10

5  6 11
Find the value of r so that the line through (10, r) and (3, 4) has
x
y
x y
a slope of -2/7
 2(7)  7(4  r )
2 4r
14  28  7r
1
7

3  10
Plug in the m and the
coordinates for the slope
Simplify
and
solve
for r
-28 -28
-14= -7r
/-7 /-7
1
2
r=2
2
4.2 Slope and Direct Variation
• Direct Variation- a proportional relationship
y = kx
*this represents a constant rate of change and k is the
constant of variation
Ex:
a. Name the constant of
variation
k = -1/2
b. Find the slope
m
20
2
1


40 4
2
c. Compare slope
and the constant of
variation. What do
you notice?
They are the
same thing
•
y
Graph a direct
Variation:
1. Write the slope as a
ratio
2. Start at the origin (0, 0)
3. Move up and across
according to the slope
4. Draw a line to connect
the ordered pairs
x
a. y = 4x
4
m
1
Up 4 and right 1
b. y = -1/3x
m
1
3
Down 1 and right 3
4.3 Slope-Intercept Form
y = mx + b
slope
y-intercept
Ex: Write an equation in slopeintercept form given slope and
the y-intercept
a.
Slope = 3
y-intercept = -5
y = 3x - 5
b.
y-intercept= (0, 6) m = 2/5
y = 2/5x + 6
Ex:
Graph from an equation:
Write an equation in
slope-intercept form given
a graph.
1. write the slope as a ratio and yintercept as an ordered pair
2. Graph the y-intercept
2  3 5
m

0  3 3
3. Use slope to find other points
and connect with a line
y-intercept=
(0, 2)
So b=2
Graph an equation given
in slope-intercept form
Ex:
y
a. y = -2/3x + 1
m = -2/3
y-int= (0, 1)
y = 5/3x + 2
b. 5x – 3y = 6
x
5x – 3y = 6
-5x
-5x
-3y = -5x + 6
/-3
y = 5/3x - 2
/-3
m = 5/3 y-int= (0, -2)
6.7 Graphing Inequalities with Two
Variables
•
•
The equation makes the line to define the boundary
The shaded region is the half-plane
1.
2.
Get the equation into slope-intercept form
List the intercept as an ordered-pair and the slope as a
ratio
Graph the intercept and use the slope to find at least 2
more points
Draw the line (dotted or solid)
Test an ordered-pair not on the line
3.
4.
5.
1.
2.
If it is true shade that side of the line
If it is false shade the other side of the line
Ex1:
< or >
 or 
Dotted Line
Solid Line
y 2x - 3
m=
2
1
b = -3 = (0, -3)
Use a solid line because it is 
Test: (0, 0)
0  2(0) – 3
0 0 – 3
0  -3
false (shade other side)
• Ex2: y – 2x < 4
y – 2x < 4
+ 2x +2x
y < 2x + 4
m=
2
Use a dotted line because it is <
1
b = 4 = (0, 4)
Test: (0, 0)
0 < 2(0) + 4
0 < 0+4
0 <4
true (shade this side)
• Ex3: 3y - 2 > -x + 7
3y – 2 > -x + 7
+2
+2
3y > -x + 9
/3
/3
/3
1
x+3
3
y>-
1
m=Use a dotted line because it is >
3
b = 3 = (0, 3)
Test: (0, 0)
1
3
0 > - (0) + 3
0 > 0+3
0 >3
false (shade other side)
4.4 Writing Equations in SlopeIntercept Form
y = mx + b
slope
y-intercept
Write an equation given slope and one point
Ex.
Write the equation of the line that passes through (1, 5) with slope 2
y = mx + b
y = 2x + b
Replace m with the slope
5 = 2(1) + b
Replace the x and y with the ordered pair coordinates
5=2+b
Solve for b (the y-intercept)
-2 -2
3=b
y = 2x + 3
Replace the numbers for
slope and the y-intercept
Write an equation given two points
Ex.
Write an equation for the line that passes through (-3, -1) and (6, -4)
y = mx + b
m
 4  1  3
1


6  3
9
3
Find slope
y = -1/3x + b
Replace m with the slope
-1 = -1/3(-3) + b
Replace the x and y with one of the ordered pair coordinates
-1 = 1 + b
-1 -1
Solve for b (the y-intercept)
-2 = b
y = -1/3x - 2
Replace the numbers for
slope and the y-intercept
4.5 Writing Equations in
Point-Slope Form
Point-Slope Form: y – y1 = m(x – x1)
Ex:
Write an equation given slope and one point
Write the equation of a line that passes through (6, -2) with slope 5
y – y1 = m(x – x1)
Replace m with the slope and y1 and x1 with the ordered pair
y – -2= 5(x – 6)
Simplify
y + 2 = 5(x – 6)
Ex: Write an equation of a horizontal line
Write the equation for the line that passes through (3, 2) and is horizontal
y – y1 = m(x – x1)
y – 2 = 0(x – 3)
y – 2 =0
y=2
Replace m with the slope and y1 and x1 with the ordered pair
Simplify
Ex: Write an equation in
Standard Form
y + 5= -5/4(x -2)
4[y + 5= -5/4(x -2)] Multiply by 4
Distribute
4y + 20= -5(x -2)
4y + 20= -5x + 10 Add 5x(move it to
the other side)
+5x
+5x
5x + 4y + 20= 10
- 20 -20
5x + 4y = -10
Subtract 20(move it
to the other side)
Ex: Write an equation in SlopeIntercept Form
y -2= ½(x+ 5)
2[y -2= ½(x+ 5)]
2y -4= x+ 5
+4= +4
2y = x+ 9
/2 /2
y = 1/2x+ 9/2
Multiply by 2
Add 4(move it to
the other side)
Divide by 2
Simplify
Simplify
Ex: Write an equation in Point-Slope Form given two points
Write the equation for the line that passes through (2, 1) (6, 4)
4 1 3
m

62 4
y - 1= 3/4(x -2)
or
y - 4= 3/4(x -6)
Find slope
Replace m with slope and y1
and x1 with one of the
ordered pairs
4.6 Statistics: Scatter Plots and
Lines of Fit
• Scatter Plot- a graph in which two sets of
data are plotted as ordered pairs in a
coordinate plane
– Used to investigate a relationship between
two quantities
Positive
Correlation
Negative
Correlation
No Correlation
• If the data points do not lie in a line, but are close to making a line
you can draw a Line of Fit
– This line describes the trend of the data (once you have this line
you can use ordered pairs from it to write an equation)
years
Ex:
Vertical
Drop
1
3
5
8
12
12
13
15
151
156
225
230
306
300
255
400
a. Make a scatter plot
500
400
300
b. Draw a line of fit. What
correlation do you find?
Positive correlation
c. Write an equation in slope-intercept
form for the line
(8, 230) (12, 306) Find slope
200
m= 19
100
1
3
5
7
9
11
13
15
Plug in an ordered
pair and slope to find
the y-intercept
y = 19x +78
4.7 Geometry : Parallel and
Perpendicular Lines
• Parallel lines- do not
• Perpendicular linesintersect and have the
make right angles and
same slope
have opposite slopes
c
a
b
Line a has slope 6/5
Line b has slope 6/5
d
Line c has slope 3/2
Line d has slope -2/3