Write down this problem on your
READY RECALL SHEET
Be prepared to explain your answer if you are called on.
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).
prepare for my Unit 11 Test by
making a 3x5 card and completing my
Practice Test.
ITEMS of BUSINESS
Test is on Monday/Tuesday. Also, it’s the last
Day 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
UNIT 11 REVIEW CONCEPTS
Write them down
on your 3x5 card.
Remember
It’s worth
An extra 1%
on your test!
POSSIBLE SOLUTIONS FOR 2 EQUATIONS
(1,3)
Intersecting Lines
Different Slope, y-int. can be
different or the same
Solve Algebraically:
You will get one solution
(X,Y)
Parallel Lines
Coincidental Lines
Same slope, different y-int.
Same line, same slope, same y-int.
Solve Algebraically:
Solve Algebraically:
Variables drop out
FALSE Statement
ex. 12 = 9
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?
𝟓
𝟑
A. y = - x - 2
𝟓
𝟑
y = - x + 10
None
B.
𝟐
y=- x
𝟕
𝟏
y= x
𝟑
One
+5
-2
C. y = 9x - 1
y = -1 + 9x
Infinite
HOW MANY SOLUTIONS?
SOLVE BY GRAPHING
{
𝟏
𝟑
y=- x+3
y = 2x – 4
Solution: (3, 2)
SOLVING BY GRAPHING
{
𝟒
𝟑
𝟒
𝟑
y=- x+5
y=- x
No Solution
{
y = 3x – 1
-3x + y = – 1
y = 3x -1
Infinite Solutions
SOLVING BY SUBSTITUTION
{
x + y = 12
x=2+y
x is equal to 2 + y
2 + y + y = 12
It is of equal value.
2 + y can replace x
We can solve this!
SUBSTITUTION
A. Solve by Substitution:
-5x + 5y = -5
x = -y + 5
{
Solution:
x, y
(3, 2)
x = -y + 5
x = -(2) + 5
x = -2 + 5
x = 3
Check:
Are these statements
TRUE?
-5(-y + 5) + 5y = -5
5y – 25 + 5y = -5
10y – 25 = -5
+25 +25
10y
= 20
10
10
y =2
-5(3) + 5(2) = -5
3 = -2 + 5
SUBSTITUTION
B. Solve by Substitution:
{
2x – 4y = 1
x = 2y + 3
2(2y + 3) – 4y = 1
4y + 6 – 4y = 1
0y + 6 = 1
6 = 1
Is this true?
NO
NO SOLUTION
SUBSTITUTION
C. Solve by Substitution:
4x – 2y = -10
y = 2x + 5
{
4x – 2(2x + 5) = -10
4x – 4x – 10 = -10
0x – 10 = -10
-10 = -10
Is this true?
YES
INFINITE SOLUTIONS
MATHICAL MATT/MAGGIE
WE ARE USING THE ELIMINATION METHOD!!
ADD THE TWO
EQUATIONS
TOGETHER & SOLVE
x + y = 30
x–y=6
2x = 36
2
2
x = 18
REPLACE x WITH 18
IN ONE OF THE ORIGINAL
EQUATIONS & SOLVE FOR Y
x = 18, y = 12
18 + y = 30
y = 30 - 18
y = 12
(18,12)
ELIMINATION
A. Solve by Elimination:
5x + 2y = 12
-5x + 4y = -66
6y = -54
6
6
y = -9
{
+
5x + 2y = 12
5x + 2(-9) = 12
5x - 18 = 12
+18 +18
5x
= 30
5
5
x =6
Solution:
x, y
(6, -9)
I WISH
3𝑥 + 3𝑦 = −21
𝐴.
3𝑥 − 2𝑦 = 14
−4𝑥 − 15𝑦 = −17
B.
−𝑥 + 5𝑦 = −13
−4𝑥 + 9𝑦 = 9
C.
𝑥 − 3𝑦 = −6
LETS PRACTICE WITH OUR
COMMUNICATORS
Example: Multiply both sides with 3
x + 2y = 7
A. Multiply both sides with 2
3x - y = 4
B. Multiply both sides with -4
x - 4y = 2
C. Oppositize both sides
x - 3y = -5
3x + 6y = 21
6x - 2y = 8
-4x + 16y = -8
-x + 3y = 5
ELIMINATION
C. Solve by Elimination:
{
( 2x – 4y = 6 )3
( -3x + 6y = -9)2
6x – 12y = 18
+ -6x + 12y = -18
0 = 0
Is this true?
Yes!
Infinite Solutions
ELIMINATION
Solve by Elimination:
(3x – 5y = -6 ) 3
-9x + 15y = 13
{
9x – 15y = -18
+ -9x + 15y = 13
0 = -5
Is this true?
No!
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 ELIM
-5x – 7 y = -2
2x – 4y = 10
SUB
x = -6y + 12
5 -3x + 4y = 1
y = 2x - 4
ELIM
SUB
15x + 3y = -2
-5x +2 y = -2
y = 3/4x -5
GRAPH
y = 5x - 8
y = mx + b
y = 6/7x +3GRAPH
y = ½ x - 12
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:
What does the solution
represent?
Y = 180 + 4x
Y = 10x
Solve the above system.
Using Substitution:
10x = 180 + 4x
-4x
-4x
6x = 180
6
6
x = 30
y = 10x
y = 10(30)
y = 300
Solution:
(30, $300)
The graph of the equations is to the right. The point of intersection is called the
31
At least ______ apple pies must be sold to make a profit.
Break-Even Point.
Break-Even
Point
Unit 11 Review
prepare for my Unit 11
Test by making a 3x5 card
and completing my Practice
Test.
Practice Test
Do the CIRCLED PROBLEMS on the
PRACTICE TEST. When you finish, you
can work on the rest of the problems.
Answers
2
7
15
17
8
19C
20
3
4
12
14
Y = 10 X 4C
Y = 5C
(10,50)
22
11 CARS
FALSE, VARIABLES DROP OUT,
NO SOLUTION
What’s your score?
21 possible
-1 = 95%
-2 = 90%
-3 = 86%
-4 = 81%
-5 = 76%
-6= 71%
-7= 67%
-8= 62%
-9 or more: F
Now, rework the ones you missed
AND finish the rest of the practice test.
Write down today’s
homework on your
CALENDAR CARD.
WHITE
Worksheet
UNIT 11
PRACTICE
TEST
