1. Analyzing Linear Equations
Chapter 4
Analyzing Linear Equations

2. 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=
Change in x x y
Ex:
Negative slope
Positive slope
Time
Walking(sec)
Distance
Walked(ft) 1 4 2 8 3 12 16 +1 4 +4
= 4 ft/sec
Rate of change = +1 +4 1
Undefined slope +1 +4
Zero slope

3. Slope- the ratio of the change in y-coordinates over the change in x-coordinates for a line
rise
Slope:
m =
y2 – y1
Slope=
run
x2 – x1
Find the slope (4, 6) (-1, 2)
Ex:
Ex: x1 y1 x2 y2
Ex:
Find the value of r so that the line through (10, r) and (3, 4) has a slope of -2/7 x1 y1 x2 y2
Simplify and solve for r
r = 2
-14= -7r
Plug in the m and the coordinates for the slope
/-7 /-7

4. 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
c. Compare slope and the constant of variation. What do you notice?
k = -1/2
They are the same thing
b. Find the slope

5. Graph a direct Variation:
Write the slope as a ratio
Start at the origin (0, 0)
Move up and across according to the slope
Draw a line to connect the ordered pairs
a. y = 4x
b. y = -1/3x
Up 4 and right 1
Down 1 and right 3

6. 4.3 Slope-Intercept Form y = mx + b slope y-intercept Ex:
Write an equation in slope-intercept form given slope and the y-intercept b.
y-intercept= (0, 6) m = 2/5 a.
Slope = 3 y-intercept = -5
y = 3x - 5
y = 2/5x + 6

7. Write an equation in slope-intercept form given a graph.
Graph from an equation:
write the slope as a ratio and y-intercept as an ordered pair
Graph the y-intercept
Use slope to find other points and connect with a line
Ex:
Write an equation in slope-intercept form given a graph.
Graph an equation given in slope-intercept form
Ex:
y-intercept= (0, 2)
So b=2
a. y = -2/3x + 1
m = -2/3 y-int= (0, 1)
y = 5/3x + 2
b. 5x – 3y = 6
5x – 3y = 6
-5x -5x
-3y = -5x + 6
/-3 /-3
y = 5/3x - 2
m = 5/3 y-int= (0, -2)

8. 6.7 Graphing Inequalities with Two Variables
The equation makes the line to define the boundary
The shaded region is the half-plane
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
If it is true shade that side of the line
If it is false shade the other side of the line

9. < or > or
Dotted Line Solid Line
Ex1: y 2x - 3
m =
b = -3 = (0, -3)
Use a solid line because it is
Test: (0, 0)
0 2(0) – 3
– 3
false (shade other side)

10. Ex2: y – 2x < 4 y – 2x < 4 + 2x +2x y < 2x + 4 m =
b = 4 = (0, 4)
Use a dotted line because it is <
Test: (0, 0)
0 < 2(0) + 4
0 <
0 < true (shade this side)

11. Ex3: 3y - 2 > -x + 7 3y – 2 > -x + 7 +2 +2 3y > -x + 9
3y > -x + 9
/3 /3 /3
y > x + 3
m = -
b = 3 = (0, 3)
Use a dotted line because it is >
Test: (0, 0)
0 > - (0) + 3
0 >
0 > false (shade other side)

12. 4.4 Writing Equations in Slope-Intercept 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)
3 = b
Replace the numbers for slope and the y-intercept
y = 2x + 3

13. Write an equation given two points
Ex.
Write an equation for the line that passes through (-3, -1) and (6, -4)
y = mx + b
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
Solve for b (the y-intercept)
-2 = b
Replace the numbers for slope and the y-intercept
y = -1/3x - 2

14. 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)
y – -2= 5(x – 6)
Replace m with the slope and y1 and x1 with the ordered pair
y + 2 = 5(x – 6)
Simplify
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)
Replace m with the slope and y1 and x1 with the ordered pair
y – 2 =0
Simplify
y = 2

15. Write an equation in Standard Form Ex:
Write an equation in Slope-Intercept Form
y + 5= -5/4(x -2)
y -2= ½(x+ 5)
4[y + 5= -5/4(x -2)]
Multiply by 4
2[y -2= ½(x+ 5)]
Multiply by 2
Distribute
4y + 20= -5(x -2)
2y -4= x+ 5
Add 4(move it to the other side)
4y + 20= -5x + 10
+4= +4
Add 5x(move it to the other side)
+5x x
2y = x+ 9
Divide by 2
5x + 4y + 20= 10
/2 /2
Subtract 20(move it to the other side)
y = 1/2x+ 9/2
Simplify
5x + 4y = -10
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)
Find slope
y - 1= 3/4(x -2)
Replace m with slope and y1 and x1 with one of the ordered pairs or
y - 4= 3/4(x -6)

16. 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

17. 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 1 3 5 8 12 13 15
Vertical Drop
151
156
225
230
306
300
255
400
Ex:
a. Make a scatter plot
b. Draw a line of fit. What correlation do you find?
Positive correlation
500
c. Write an equation in slope-intercept form for the line
400
300
(8, 230) (12, 306) Find slope
200
m= 19
100
y = 19x +78
Plug in an ordered pair and slope to find the y-intercept 1 3 5 7 9 11 13 15

18. 4.7 Geometry : Parallel and Perpendicular Lines
Parallel lines- do not intersect and have the same slope
Perpendicular lines- make right angles and have opposite slopes c a
Line a has slope 6/5
Line b has slope 6/5 b d
Line c has slope 3/2
Line d has slope -2/3