Chapter 2: Systems of linear
equations and inequalities
Section 2-2: Solving systems of
Equations in three variables
Three variables
•
For a system of equations in three variables, the graph of each
equation is a plane in space rather than a line.
•
It is easier to use the algebraic techniques of elimination and
substitution rather than attempt to graph a system of equations in
three variables.
Example # 1
•
Solve by elimination
5x-2y+z=11
9x+7z=11
Eliminate
the Z
2x+y+3z=0 Eliminate the y
6x-2y-2z=16
10x+4z=16
34x=68
x=2
Now go backwards. X=2 Therefore 10(2)+4z=16 which means Z=-1.
And if x=2 Z=-1 then 2(2)=y+3(-1)=0 and that means y=-1. So the
solution is (2,-1,-1)
Notice after I get the x, I can put it into either of the two equations
9x+7z=11 or 10x+4z=15. And once I have the z, I can put the x and
z into any of the original three equations to get the y. After I have the
solution, I would make sure it works in all three of the original
equations.
Example #2
• Solve by substitution
3y=-9z
4x+2y-2z=0
-3x-y+4z=-2
We can solve the first equation for y. y=-3z. Then substitute this in to
the other two equations.
4x +2(-3z)-2z=0
4x-8z=0
12x-24z=0
z=-2
-3x-(-3z)+4z=-2
-3x+7z=-2
-12x+28z=-8
Now working backwards when z=-2 then 4x-8(-2)=0 means x=-4. Then
putting z=-2 and x=-4 into one of the original equations 4(-4)+2y-2(2)=0 means y=6. Therefore the solution to the system is (-4,6,-2)
Math and the Egyptians
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How are symbols used to identify something? The Egyptians had 7
symbols to represent numbers.
What base did the Egyptians use. How was this different from the
Mayans and Babylonians ?
What is the binary system and where is it used?
What symbols were used for 1, 10, 100, 1000,10000, 100000,
1000000?
Did the Egyptians have the zero as a place value?
What is doubling and addition in Egyptian math?
How did the Egyptians work with division?
How did Egyptians work with fractions?
What is the only exception to the unit fraction rule?
What is the Greedy Method? Who created it?
What is the largest building on the planet?