10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Warm­up:
2. Solve the system.
­x + 6y = 8
2x ­ 12y = ­14
1. Solve the system. 2x ­ y = 11
x + 2y = ­7
3. Determine whether the point (1, 3) is a solution of each equation in the system.
2x + 5y = 17
­4x + 3y = 5
Oct 19­5:24 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Dec 1­12:20 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Dec 1­12:22 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Dec 1­12:23 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
10.1 Systems of Linear Equations: Substitution and Elimination
Day 2
Objectives :
1. Identify inconsistent systems of equations containing three variables. 2. Solve systems of three equations containing three variables.
3. Express the solution of a system of dependent equations containing three variables. Nov 9­12:04 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Systems of Three Equations Containing Three Variables
Example:
x ­ 3y + 3z = ­4
2x + 3y ­ z = 15
4x ­ 3y ­ z = 19
The graph of any equation of the form Ax + By + Cz = D is a plane.
The solutions of a three­variable system can be shown graphically as the intersection of planes.
Oct 19­5:16 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Systems of Three Equations Containing Three Variables
One Solution
The planes intersect at one common point.
Many Solutions
No Solution
No point lies in all The planes intersect at all the points along three planes.
a common line.
Solution
System is consistent and equations are independent.
System is consistent and equations are dependent.
System is inconsistent.
Oct 19­5:16 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Solve the system by elimination. x ­ 3y + 3z = ­4
2x + 3y ­ z = 15
4x ­ 3y ­ z = 19
Step 1: Pair the equations to eliminate one of the variables.
Step 2: Write the two new equations as a system with two variables.
Step 3: Solve for the two variables in the new system.
Oct 19­5:20 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Solve the system by elimination. x ­ 3y + 3z = ­4
2x + 3y ­ z = 15
4x ­ 3y ­ z = 19
Step 4: Find the last variable by back­substitution.
Step 5: State your results and check your answer.
(5, 1, ­2)
Oct 19­5:20 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Solve the system by elimination!
2x + y ­ z = ­2
x + 2y ­ z = ­9
x ­ 4y + z = 1
no solution
inconsistent
Oct 19­5:20 PM
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10.1 Day 2 Systems of Linear Equations 2010 Try Another!
December 01, 2010
Solve the system by elimination.
2x + y ­ z = 5
3x ­ y + 2z = ­1
x ­ y ­ z = 0
(1, 2, ­1)
Oct 19­5:20 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Solve the system by elimination.
x ­ 2y ­ z = 8
2x ­ 3y + z = 23
4x ­ 5y + 5z = 53
many
solutions
Oct 19­6:15 PM
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10.1 Day 2 Systems of Linear Equations 2010 This one's a little harder!
December 01, 2010
Solve the system by elimination.
2x + y ­ z = 5
x + 4y + 2z = 16
15x + 6y ­ 2z = 12
(­2, 6, ­3)
Oct 19­6:15 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Theater Revenues: A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $50, main seats for $35, and balcony seats for $25. If all the seats are sold, the gross revenue to the theater is $17,100. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $14,600. How many are there of each kind of seat?
100 orchestra
210 main
190 balcony
Nov 28­1:36 PM
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10.1 Day 2 Systems of Linear Equations 2010 December 01, 2010
Homework
page 738
(15 ­ 16,
41 ­ 43,
46, 48, 76)
Nov 28­12:53 PM
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