Download CD = infinitely many solutions

ALGEBRA
Quick Chart – Systems of Equations
MEMORIZE THIS CHART! QUIZ TOMORROW!
Type
Slopes
Starting points
(SP)
Solution(s)
Graphs look like
Example graph
Consistent
Independent
y = 3x – 1
y = 2x + 5
2x – 5y = 7
2x + 5y = -7
Consistent
Dependent
y = 3x + 4
3x – y = - 4
y = 2x – 3
2y – 4x = 6
Inconsistent
y = 3x – 7
y = 3x + 2
2x – 5y = 8
5y – 2x = 12
CI = consistent independent
CD = consistent dependent
INC = inconsistent
CI = one solution
CI = Different slopes
CD = infinitely many solutions
CD = Same Slope, Same SP
INC = no solutions
INC = Same Slope, Diff SPs
1
HOW TO SOLVE BY GRAPHING:
1.
2.
3.
Example:
1.
y=-x+1
y = 2x + 4
Solution
3.
CI = one solution
Graph both of the equations.
CD = infinitely many solutions
Find the point(s) of intersection.
INC = no solutions
a. One-point –solution is an ordered pair
b. ALL points-same line-solution is INFINITE NUMBER of solutions
c. No points-parallel lines-solution is NO SOLUTION
CI = Different slopes
CHECK YOUR SOLUTION in BOTH equations!
x – 2y = 4
y=½x–4
Solution
CD = Same Slope, Same SP
INC = Same Slope, Diff SPs
2.
y = 3x – 3
y–x=1
Solution
4.
3x + y = -5
y = -3x – 5
Solution
1. When do you think you will get “one solution”?
2. When do you think you will get “infinitely many solutions”?
3. When do you think you will get “no solution”?
2
Go to the first link in canvas. The videos are all in a row in Khan Academy.
Example 1: (this will be an example of an INCONSISTENT
Example 2:
system)
y  4 x  17.5
y
1
x  100
4
1
y
x  120
4
y  2 x  6.5
Example 3: (only do the substitution, you do not need to
do the graph portion)
The Substitution Method – TRY ON YOUR OWN! 
9 x  3 y  15
yx5
Substitution Method 2 - TRY ON YOUR OWN! 
y  5 x  8
10 x  2 y  2
2y  x  7
x  y4
Practice using substitution for systems - TRY ON YOUR
OWN! 
3x  4 y  2
y  2x  5
3
1.
2.
3.
4.
5.
6.
Line up the variables.
Make sure to find "opposites" for one of the variables.
a. If you do not have opposites right away:
i. Look to find a common multiple for ONE of the variables. Make sure that they will be
OPPOSITES! You may have to multiply a negative through to get opposites.
ii. Multiply the entire equation by the number that will get you the LCM of the coefficient.
iii. You may have to multiply the other entire equation by the number that will get you the LCM.
Add the two equations together.
Solve for the variable. You should only have one variable left.
Plug your answer for number 4 into one of the equations and solve for the missing variable.
Check your solution by plugging it into the other equation.
SPECIAL CASES:
1.
2.
If you get two numbers that are equal and no variables left, your answer is infinitely many solutions.
8 = 8 means your answer is infinitely many solutions
If you get two numbers that are NOT equal and no variables left, your answer is no solution.
9 = 7 means your answer is NO SOLUTION
4 x  2 y  12
x  y  11
4 x  8 y  24
2 x  y  19
2 x  9 y  25
6 x  6 y  6
4 x  9 y  23
6 x  3 y  12
4
5x  y  9
3 x  7 y  16
10 x  7 y  18
9 x  5 y  16
16 x  10 y  10
4 x  15 y  17
8 x  6 y  6
 x  5 y  13
7 x  8 y  9
-4x - 2 y = 14
-10x + 7 y = -25
4 x  9 y  22
5
In-Class (Special Cases)
3.6 x  0.6 y  3
3x  y  3
0.9 x  1.4 y  8
3x  y  3
4x  4  y
x  3y  6
4 x  y   4
x  3y  6
x  3y  9
2x  y  6
 x  3 y  9
2x  y  6
6
You are going to watch 4 videos. You need to write out what you think is the important points of the
videos.
Video 1 – What is a relation?
Video 2 – What is a function? Make sure to state what domain and range represent.
Video 3 – How can you tell if a relation is not a function?
Video 4 – Vertical Line Test
In-Class Practice
A student finished the following worksheet. State if his/her answers are correct. If they are wrong, state the error.
7
Directions: Solve the following problems. Match that answer to the correct letter of the
alphabet. Enter that letter of the alphabet on the blank corresponding to the problem
number.
A
9
B
0
C
-1
D
-16
E
N
-7
O
4
P
5
Q
7
R
8
Simplify:
1. f(x) = 2x – 1
2. f(x) = x2 – 3x –1
3. f(x) = 2x + 5
4. f(x) = -2x2 – 5
5. f(x) = x + 5
6. f(x) = 6x2 + 2x
7. f(x) =
1
x + 2x
4
8. f(x) = 4x – 5
18
F
16
G
-2
H
-4
I
S
23
T
-5
U
-8
V
15
3
J
2
K
-9
W
-23
L
X
11
1
M
-3
Y
42
Z
-18
Find f(5).
Find f(3).
Find f(0).
Find f(-1).
Find f(-7).
Find f(1).
9. f(x) = x3 – 2x – 1
10. f(x) = x4 + 2x2 – 1
11. f(x) = -4x – 8
12. f(x) = 2x – 10
13. f(x) = x3 – 2x2 + x + 5
14. f(x) = x2 – 21
Find f(-2).
Find f(2).
Find f(-1).
Find f(1).
Find f(-1).
Find f(5).
Find f(8).
15. f(x) = (x – 2)2
Find f(-2).
Find f(2).
___ ___ ___ ___ ___ ___ ___ ___ ___
15 12 4
2
9
8 14 4 10
___ ___ ___ ___ ___ ___ ___ ___
10 9
6
1
8
5 11 9
___ ___ ___ ___
3
1 10 10
___ ___ ___ ___
13
8
4
7
___ ___ ___
9
11 7
___ ___ ___ ___
9
7 10 9
8
Step 1: Graph the first inequality.
Step 2: SHADE the first inequality. Make sure to check if the line is dotted or solid.
Step 3: Graph the second inequality.
Step 4: Shade the second inequality. Make sure to check if the line is dotted or
solid.
Step 5: Find the part where they overlap (where the graph goes plaid). The
overlapped area is your solution. Make sure to delete the sections that do NOT
overlap!
9
Ex. 1 – 2 Jose's Coffee Shop makes a blend that is a mixture of two types of coffee. Type A coffee costs Jose $5.65 per
pound, and type B coffee costs $4.50 per pound. This month's blend used twice as many pounds of type B coffee as type
A, for a total cost of $468.80. How many pounds of type A coffee were used?
Ex. 1 – 2 Kala's Coffee Shop makes a blend that is a mixture of two types of coffee. Type A coffee costs Kala $5.60 per
10
pound, and type B coffee costs $4.25 per pound. This month, Kala made 147 pounds of the blend, for a total cost
of $755.70. How many pounds of type A coffee did she use?
Ex. 1 – 2 At the movie theatre, child admission is $5.50 and adult admission is $8.70. On Wednesday, 183 tickets were
sold for total sales of $1275.30. How many child tickets were sold that day?
Ex. 1 – 2 A Web music store offers two versions of a popular song. The size of the standard version is 2.5 megabytes
(MB). The size of the high-quality version is 4.6 MB. Yesterday, there were 1470 downloads of the song, for a total
download size of 4977 MB. How many downloads of the high-quality version were there?
Ex. 3 – 4 Ryan invested his savings in two investment funds. The amount he invested in Fund A was $7000 less than the
amount he invested in Fund B. Fund A returned a 7% profit and Fund B returned a 3% profit. How much did he invest in
Fund B, if the total profit from the two funds together was $1110?
Amt. invested
Profit %
Profit amt.
Fund A
Fund B
Total
Ex. 3 – 4 In the lab, Kaitlin has two solutions that contain alcohol and is mixing them with each other. Solution A is 20%
alcohol and Solution B is 6% alcohol. She uses 1000 milliliters of Solution A. How many milliliters of Solution B does she
use, if the resulting mixture is a 16% alcohol solution?
Vol. of Sol.
% alc.
Vol. alc.
Sol. A
Sol. B
Mixture
Ex. 3 – 4 Brian invested his savings in two investment funds. The $2000 that he invested in Fund A returned a 5% profit.
The amount that he invested in Fund B returned a 2% profit. How much did he invest in Fund B, if both funds together
returned a 3% profit?
Amt. invested
Profit %
Profit amt.
Fund A
Fund B
Total
11
Ex. 5 – 8 Kaitlin drove to the mountains last weekend. There was heavy traffic on the way there, and the trip took 9
hours. When Kaitlin drove home, there was no traffic and the trip only took 4 hours. If her average rate was 40 miles per
hour faster on the trip home, how far away does Kaitlin live from the mountains?
Rate (mph)
Time (hrs.)
Distance (mi.)
Driving to Mountains
Driving Home
Ex. 5 – 8 Two cars leave towns 760 kilometers apart at the same time and travel toward each other. One car's rate
is 18 kilometers per hour less than the other's. If they meet in 4 hours, what is the rate of the slower car?
Rate (mph)
Time (hrs.)
Distance (mi.)
Slower car
Faster car
Ex. 5 – 8 Two trains leave stations 336 miles apart at the same time and travel toward each other. One train travels
at 65 miles per hour while the other travels at 75 miles per hour. How long will it take for the two trains to meet?
Rate (mph)
Time (hrs.)
Distance (mi.)
Train #1
Train #2
Ex. 5 – 8 Two cars leave towns 570 kilometers apart at the same time and travel toward each other. One car's rate is
18 kilometers per hour more than the other's. If they meet in 3 hours, what is the rate of the faster car?
Rate (mph)
Time (hrs.)
Distance (mi.)
Faster Car
Slower Car
12