Today’s Agenda
 Attendance

/ Announcements
 Collect Projects
 Return Exams
 Sections 6.1, 6.2
 Quiz Friday
Exam Schedule
 Exam

4 (Ch 6,7)
Fri 11/15
 Exam 5 (Ch 10)
Thur 12/5
 Final Exam (All)
Thur 12/12
Review: Graphing/Equations of
Lines

Sections 2.1, 2.2
Systems of Linear Equations
3x y 9
2x y 1
A solution of a system of linear
equations is an ordered pair that
satisfies bot...
Solve by Graphing
3x y 9
2x y 1
So, the
solution of
the system
is (2, -3)
Solve by Graphing
2x 4 y 8
y

1
2

x 2

So, the
solution of
the system
is ?
Solve by Graphing
x 2y
2x 4 y

3
6

So, the
solution of
the system
is ?
Solve by Graphing
Solving Systems Algebraically
The Substitution Method
• Works well when it is easy to solve
for one of the variables
• Sub...
The Substitution Method
3x y 6
6x 5 y
23

• What does this tell us about the
graphs of the system?
The Substitution Method
3x y 9
2 y 6x 7

• What does this tell us about the
graphs of the system?
The Substitution Method
4x y
8x 2 y

2
4

• What does this tell us about the
graphs of the system?
The Substitution Method
x 2y
2x y

3
5

• What does this tell us about the
graphs of the system?
Solving Systems Algebraically
The Elimination Method
• We can multiply/divide any linear
equation.
• We can also add/subtr...
The Elimination Method
5x y 4
3x 2 y 1
Addition Method
- Want to get
“opposite
coefficients”
- Then add
equations to
elimi...
The Elimination Method
2x 3y 2
2 y 3 x 12

Want to rewrite to line up like
terms
The Elimination Method
2 x 5 y 14
7x 6 y
2
Applications
The sum of two numbers is 70 and
their difference is 24. Find the two
numbers.
Solving Larger Systems using
Substitution
• Use algebra to isolate one of the
variables, substitute into the other
two.
• ...
The Substitution Method
2x y z

2

x 3 y 2z 1
x y z

2
The Substitution Method
2x 3 y

2

x y 4z

16

3x 2 y z

7
The Substitution Method
x 2y z 5
2x y z
2x 4 y 2z

2
2
Classwork / Homework

• Page 317
•1 – 15 odd, 19
• Page 329
•9, 39, 41, 45
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Lecture 6.1 6.2 bt

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Lecture 6.1 6.2 bt

  1. 1. Today’s Agenda  Attendance / Announcements  Collect Projects  Return Exams  Sections 6.1, 6.2  Quiz Friday
  2. 2. Exam Schedule  Exam 4 (Ch 6,7) Fri 11/15  Exam 5 (Ch 10) Thur 12/5  Final Exam (All) Thur 12/12
  3. 3. Review: Graphing/Equations of Lines Sections 2.1, 2.2
  4. 4. Systems of Linear Equations 3x y 9 2x y 1 A solution of a system of linear equations is an ordered pair that satisfies both equations.
  5. 5. Solve by Graphing 3x y 9 2x y 1 So, the solution of the system is (2, -3)
  6. 6. Solve by Graphing 2x 4 y 8 y 1 2 x 2 So, the solution of the system is ?
  7. 7. Solve by Graphing x 2y 2x 4 y 3 6 So, the solution of the system is ?
  8. 8. Solve by Graphing
  9. 9. Solving Systems Algebraically The Substitution Method • Works well when it is easy to solve for one of the variables • Substitute expression into other equation, Solve. • Plug solved value back into either equation to find solution. • Check the solution in both equations
  10. 10. The Substitution Method 3x y 6 6x 5 y 23 • What does this tell us about the graphs of the system?
  11. 11. The Substitution Method 3x y 9 2 y 6x 7 • What does this tell us about the graphs of the system?
  12. 12. The Substitution Method 4x y 8x 2 y 2 4 • What does this tell us about the graphs of the system?
  13. 13. The Substitution Method x 2y 2x y 3 5 • What does this tell us about the graphs of the system?
  14. 14. Solving Systems Algebraically The Elimination Method • We can multiply/divide any linear equation. • We can also add/subtract linear equations together. • Use algebra to eliminate one of the variables, solve for the other. • Check the solution in both equations
  15. 15. The Elimination Method 5x y 4 3x 2 y 1 Addition Method - Want to get “opposite coefficients” - Then add equations to eliminate variable Subtraction Method - Want to get “same coefficients” - Then subtract equations to eliminate variable
  16. 16. The Elimination Method 2x 3y 2 2 y 3 x 12 Want to rewrite to line up like terms
  17. 17. The Elimination Method 2 x 5 y 14 7x 6 y 2
  18. 18. Applications The sum of two numbers is 70 and their difference is 24. Find the two numbers.
  19. 19. Solving Larger Systems using Substitution • Use algebra to isolate one of the variables, substitute into the other two. • Solve the two variable system, use Substitution find the first variable • Check solution in all three equations
  20. 20. The Substitution Method 2x y z 2 x 3 y 2z 1 x y z 2
  21. 21. The Substitution Method 2x 3 y 2 x y 4z 16 3x 2 y z 7
  22. 22. The Substitution Method x 2y z 5 2x y z 2x 4 y 2z 2 2
  23. 23. Classwork / Homework • Page 317 •1 – 15 odd, 19 • Page 329 •9, 39, 41, 45

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