Tate and Anna are selling pies for a school fundraiser. Customers can buy apple pies and blackberry pies. Tate sold 7 apple pies and 8 blackberry pies for a total of $155. Anna sold 2 apple pies and 4 blackberry pies for a total of $70. Write a system of equations to represent this situation (do not need to solve). i-can Write down this problem on your READY RECALL SHEET Be prepared to explain your answer if you are called on. prepare for my Unit 11 Test making a 3x5 card and completing my Practice Test.
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Tate and Anna are selling pies for a school fundraiser. Customers can buy apple pies and blackberry pies. Tate sold 7 apple pies and 8 blackberry pies for a total of $155. Anna sold 2 apple pies and 4 blackberry pies for a total of $70. Write a system of equations to represent this situation (do not need to solve).
i-can
Write down this problem on yourREADY RECALL SHEET
Be prepared to explain your answer if you are called on.
prepare for my Unit 11 Test bymaking a 3x5 card and completing myPractice Test.
ITEMS of BUSINESS
Test is on Monday/Tuesday. Also, it’s the lastDay to turn in any absent/missing work.
SAGE test is May 7-12. BE HERE.
Questions on Homework?then Trade-n-Grade
SYSTEMS OF EQUATIONS
Unit 11 Review
Write them down on your 3x5 card.
UNIT 11 REVIEW CONCEPTS
Remember It’s worth
An extra 1% on your test!
POSSIBLE SOLUTIONS FOR 2 EQUATIONS
One Solution No SolutionInfinitely
Many Solutions
(1,3)
Intersecting LinesDifferent Slope, y-int. can be
different or the same
Solve Algebraically: You will get one solution
(X,Y)
Parallel LinesSame slope, different y-int.
Solve Algebraically: Variables drop out FALSE Statement
ex. 12 = 9
Coincidental LinesSame line, same slope, same y-int.
Solve Algebraically: Variables drop out TRUE Statement
ex. 22 = 22
LETS TRY A FEW PROBLEMS
Given the following systems, can You determine how many solutions
(One, None, or Infinite) w/o graphing?A. y = - x - 2 y = - x + 10
B. y = - x + 5 y = x - 2
C. y = 9x - 1 y = -1 + 9x
InfiniteOneNone
HOW MANY SOLUTIONS?
SOLVE BY GRAPHING
y = - x + 3y = 2x – 4 {
Solution: (3, 2)
SOLVING BY GRAPHINGy = - x + 5 y = - x
y = 3x – 1 -3x + y = – 1{{
No Solution Infinite Solutions
y = 3x -1
SOLVING BY SUBSTITUTION
x + y = 12
x = 2 + y
2 + yx is equal to 2 + y
{2 + y can replace xIt is of equal value.
2 + y + y = 12 We can solve this!
A. Solve by Substitution:-5x + 5y = -5 x = -y + 5{
-5(-y + 5) + 5y = -5 5y – 25 + 5y
10y – 25 = -5 +25 +25
10y = 20 10 10
y = 2
= -5
x = -y + 5x = -(2) + 5
x = 3
Solution: (3, 2) x, y
x = -2 + 5
SUBS
TITU
TIO
N
-5(3) + 5(2) = -5
3 = -2 + 5Check:Are these statements TRUE?
B. Solve by Substitution:
2(2y + 3) – 4y = 1
2x – 4y = 1 x = 2y + 3{ 4y + 6 – 4y
0y + 6 = 1 6 = 1
Is this true?
= 1
NOSU
BSTI
TUTI
ON
NO SOLUTION
C. Solve by Substitution:
4x – 2(2x + 5) = -10
4x – 2y = -10y = 2x + 5 {
4x – 4x – 10 0x – 10 = -10
-10 = -10Is this true?
= -10
YESSUBS
TITU
TIO
N
INFINITE SOLUTIONS
x + y = 30x – y = 6 ADD THE TWO
EQUATIONS TOGETHER & SOLVE
2x = 362 2x = 18
REPLACE x WITH 18IN ONE OF THE ORIGINAL
EQUATIONS & SOLVE FOR Y
18 + y = 30y = 30 - 18y = 12
x = 18, y = 12 (18,12)
MATHICAL MATT/MAGGIE
ARE YOUMATHICAL
NOW? Show your PartnerWE ARE USING THE ELIMINATION METHOD!!
A. Solve by Elimination:{ 5x + 2y = 12
-5x + 4y = -66+ 6y = -54 6 6y = -9
5x + 2y = 12 5x + 2(-9) = 12
5x - 18 = 12+18 +18
5x = 305 5
x = 6
Solution: (6, -9)
x, y
ELIM
INAT
ION
I WISH
𝐴 .3 𝑥+3 𝑦=−213 𝑥−2 𝑦=14
B.
C .−4 𝑥+9 𝑦=9𝑥−3 𝑦=−6
LETS PRACTICE WITH OUR COMMUNICATORS
Example: Multiply both sides with 3 x + 2y = 7 3x + 6y = 21
A. Multiply both sides with 2 3x - y = 4 6x - 2y = 8B. Multiply both sides with -4 x - 4y = 2 -4x + 16y = -8
C. Oppositize both sides x - 3y = -5 -x + 3y = 5
C. Solve by Elimination:
Is this true?Yes!
– 12y
{ 2x – 4y = 6
-3x + 6y = -9 6x -6x+
0 = 0
( )3
= 18
( )2
+ 12y = -18
Infinite Solutions
ELIM
INAT
ION
Solve by Elimination:
– 15y
{ 3x – 5y = -6 -9x + 15y = 13
9x-9x + 15y = 13+
0 = -5Is this true?
( ) 3
= -18
No!ELIM
INAT
ION
NO SOLUTION
What does it Look Like?
ONE SOLUTION: (X,Y) ONE SOLUTION: (X,Y)
FALSE STATEMENT FALSE STATEMENT
TRUE STATEMENT TRUE STATEMENT
WHICH METHOD SHOULD I USE?
5x + 8y = 3-5x – 7 y = -2
y = 2x - 4-5x +2 y = -2
y = 3/4x -5y = 5x - 8
2x – 4y = 10x = -6y + 12
-3x + 4y = 115x + 3y = -2
y = 6/7x +3y = ½ x - 12
SUB
SUB
ELIM
ELIM
GRAPH GRAPH
5
y = mx + b y = mx + b
STORY PROBLEM The cost of making apple pies includes the booth rental of $180
and $4 per pie. We are charging $10 per pie. Write two equations, one representing the Production Cost and the other representing the Income. Assign y (dependent variable) to the Production Cost/Income and x (independent variable) to the number of pies sold.
Production Cost: Income from Sales:Solve the above system.
What does the solution represent?
The graph of the equations is to the right. The point of intersection is called the Break-Even Point. At least ______ apple pies must be sold to make a profit.
Y = 180 + 4xY = 10x
Using Substitution:
10x
10x = 180 + 4x-4x -4x
6x = 180 6 6 x = 30
y = 10x y = 10(30) y = 300
Solution: (30,
$300)
Break-Even Point
31
i-can prepare for my Unit 11 Test by making a 3x5 card and completing my Practice Test.
Unit 11 Review
Do the CIRCLED PROBLEMS on the PRACTICE TEST. When you finish, you can work on the rest of the problems.
Practice Test
Answ
ers2
3
4
7
8
12
14
15
17
19C
20
22
Y = 10 X 4C Y = 5C (10,50) 11 CARS
FALSE, VARIABLES DROP OUT, NO SOLUTION
21 possible
Now, rework the ones you missed AND finish the rest of the practice test.