1. Basic Equations

Solving equation = finding value(s) for the
unknown that make the equation true

Add, subtract, multiply, divide, raise to powers,
take roots, find least common denominators to
isolate the unknown
1
Example 1
5x + 3 = 4 + 9x
2
Example 2
3
1
4

 2
x  2 x  4 x  2x  8
3
Example 2: Check

Watch out for extraneous solutions!!!
3
1
4

 2
x  2 x  4 x  2x  8
Check
by putting it into the original equation.
4
Example 3
2
1
1


x x  1 x( x  1)
5
Example 3: Check

Check for extraneous solutions!!!
2
1
1


x x  1 x( x  1)
Check
by putting it into the original equation.
6
Example 4
ax  b
4
cx  d
7
Fractional Powers
something 


numerator
denominator
Even numerator: plus/minus possible answers
Odd numerator: only one possible answer
x
4/3
( x  1)
1
3/ 2
8
8
2. Modeling With Equations





Draw a picture if possible
Identify the quantity that you want to find
List known and unknown quantities
Set up model/equation relating quantities
Solve equation and check
9
Example 1
A woman earns 15% more than her husband.
Together, they make $69,875. How much
does the husband make?
10
Example 2
A man is four times as old as his daughter.
In 8 years, he will be 3 times as old as she.
How old is the daughter now?
11
Example 2 (continued)
Now
In 8 years
Daughter
Father
What equation relates these quantities?
Solve equation for x.
12
Example 3
What quantity of silver should be added to 90
grams of an alloy containing 40% silver and
60% gold in order to produce an alloy with
50% silver?
13
Example 3 (continued)
14
Example 4
Two cyclists, 90 miles apart, start riding toward
each other at the same time. One cycles three times
as fast as the other. If they meet 2 1/2 hours later, at
what average speed is each cyclist traveling?
15
Example 4 (continued)
Need an equation relating distance, time and speed
16
Example 5
Bob and Jim are mowing Bob’s lawn with their mowers. If they
work together, the lawn is done in 30 minutes. If Bob works
alone, it will take 25% less time than if Jim works alone. How
much time is required to finish the lawn by each person alone?
17
Example 5 (continued)
18
3. Coordinate Plane




Ordered pairs of numbers form a two-dimensional region
x-axis: horizontal line
y-axis: vertical line
Axes intersect at origin O (0,0) and divide plane into 4 parts
y
x
19
Distance Formula
y
A
B
x
Point A has coordinates:
Point B has coordinates:
Vertical distance, v, is
Horizontal distance, h, is
20
Distance Formula(continued)
y
A
B
x
21
Example 1
Find the distance between (5, 4) and (2, -1).
First, draw both points and make a guess.
22
Example 2
Find the point on the y-axis that is equidistant
from the points (1, 2) and (4, -2).
First, draw both points and make a guess.
Whatever the point, need the distance from it to point 1 to be the
same as the distance from it to point 2. Also, we know that any
point on the y-axis has
23
Example 2(continued)
(1,2)
(4,-2)
24
Midpoint Formula
Goal: Find the point that is located halfway
between two points.
( x1 , y1 )
( x2 , y2 )
Midpoint:
25
Example 3
Find the midpoint for the two points: (-2, 5) and (6, 1).
Midpoint:
26
Example 4
Find the point that is ¼ of the distance
from (2, 7) to (8, 3).
7
3
2
8
27
Example 5
Where should point S be located so that PQRS
is a parallelogram?
R(11,7)
Q(-2,6)
P(-5,-4)
S(x,y)
28
4. Graphs of Equations




Equations in two variables can be represented by a graph.
Each ordered pair (x,y) that makes the equation true is a
point on the graph.
Graph equation by plotting points and then connecting the
points with smooth curves.
Use curves except for linear equations
29
Example 1
5 x  y  12
Create a table of points:
x y=-5x+12
-2
-1
0
1
2
y
24
8
x
-1
2
30
Example 2
Create a table of points:
x y  x3  1
-2
-1
0
1
2
y  x 1
3
y
6
2
-1
x
2
-6
31
Intercepts
Where the graph crosses the axes
x-intercept: cross the x-axis => when y = 0
y-intercept: cross the y-axis => when x = 0
32
Example 3
Find the intercepts for the following equation.
y  x5
y-intercepts:
x-intercepts:
x-intercepts:
y-intercepts:
33
Example 4
Find the intercepts for the following equation.
y  3xy  x  4
2
y-intercepts:
x-intercepts:
x-intercepts:
y-intercepts:
34
Symmetry
If the graph has a mirror-image over x-axis, y-axis, or origin.
y-axis symmetry:
x-axis symmetry:
symmetry over the y-axis(x=0) symmetry over the x-axis(y=0)
35
Symmetry - continued
origin symmetry:
symmetry over the origin (0, 0)
36
Example 5
y  xx
3
Try y-axis symmetry: -x for x
Try origin symmetry:
-x for x, -y for y
Try x-axis symmetry: -y for y
37
Example 6
y  x 3
Try y-axis symmetry: -x for x
4
Try origin symmetry:
-x for x, -y for y
Try x-axis symmetry: -y for y
38
5. Lines




Linear equations also known as lines.
Each line is defined by: intercepts and slope
Slope is the change in y over the change in x
rise over run
39
Slope
y
(5,6)
(1,3)
x
40
Forms of Line
General equation:
Slope-intercept equation:
Point-slope equation:
41
Example 1
Given two points, (-2, 9) and (4, 1), on a line,
find the equation of the line.
9
-2
4
42
Example 2
Find the equation of the line given
the x-int of -8 and y-int of 3.
3
-8
43
Example 3
Find the equation of the line that passes through the
point (11, 4) and is parallel to the line y = 3x + 5.
8
4
10
44
Example 4
Find the equation of the line that passes through the
point (4, -1) and is perpendicular to the line y = -6x + 7.
45
Application
Biologists have observed a linear relationship between the
chirping rate of crickets and temperature. With 120 chirps per
minute the temp was 70 degrees Fahrenheit and with 168 chirps
per minute the temp was 80 degrees Fahrenheit.
Find the linear equation that relates temp and number of chirps
per minute.
T
90
50
10
n
40
120
200