systems of equations with gaussian elimination. system of equations find all pairs of x and y values...
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Systems of equations
With Gaussian elimination
System of equations
Find all pairs of x and y values that make the equations true.
System of equations
Swap the order of the rowsR1 <-> R2
System of equations
Multiply a row by a number-4*R1 R1
System of equations
Add a row to another rowR1 + R2 R2
System of equations
Multiply a row by a number-¼*R1 R1
The add-multiply shortcut
Multiply a row by a number and add it to another row
-4*R1 + R2 R2
Row operations
• Swap rows R1<-> R2• Multiply a row by a number k*R1 R1• Add rows together R1 + R2 R2• Multiply-add shortcut k*R1 + R2 R2
Gaussian Elimination
• A method that you can use to solve ANY system of equations (no matter how big), using only two rules.
• Multiply a row by a number k*R1 R1• Multiply-add shortcut k*R1 + R2 R2
How to solve a system of (any number of) linear equations
Method: Gaussian Elimination• Today’s fun irrelevant fact: Gauß is my great-
great-great-great-great-great-great-grand-advisor
• Gauß Gerling Plucker Klein Bocher Ford Engen Steffe Thompson Castillo-Garsow
The method
• Write equations in standard form• Use multiply to get 1x in the top equation• Use multiply-add to get 0x in all other
equations.• Use multiply to get 1y in the second equation• Use multiply-add to get 0y in all other
equations.• Repeat for all variables.
Gaussian Elimination
• Get your system in standard form(All the variables on one side, all the constants on the other)4x + 8y - 4z = 82x + 3y + 4z = 45x + 8y + 1z = 7
Gaussian Elimination
• Use multiply to get 1x in the top equation4x + 8y - 4z = 8 (1/4) * R1 --> R12x + 3y + 4z = 45x + 8y + 1z = 7
1x + 2y - 1z = 22x + 3y + 4z = 45x + 8y + 1z = 7
Gaussian Elimination
• Use multiply-add to get 0xs everywhere else1x + 2y - 1z = 22x + 3y + 4z = 4 -2 * R1 + R2 --> R25x + 8y + 1z = 7 -5 * R1 + R3 --> R3
1x + 2y - 1z = 20x - 1y + 6z = 00x - 2y + 6z = -3
Gaussian Elimination
• Use multiply to get 1y in the second equation1x + 2y - 1z = 20x - 1y + 6z = 0 -1 * R2 --> R20x - 2y + 6z = -3
1x + 2y - 1z = 20x + 1y - 6z = 00x - 2y + 6z = -3
Gaussian Elimination
• Use multiply-add to get 0ys in all other equations• You can do all of these now, but I’m going to put one off for
later.
1x + 2y - 1z = 20x + 1y - 6z = 00x - 2y + 6z = -3 2 * R2 + R3 --> R3
1x + 2y - 1z = 20x + 1y - 6z = 00x + 0y - 6z = -3
Gaussian Elimination
• Use multiply to get 1z in the third equation1x + 2y - 1z = 20x + 1y - 6z = 00x + 0y - 6z = -3 (-1/6) * R3 --> R3
1x + 2y - 1z = 20x + 1y - 6z = 00x + 0y + 1z = 0.5
Gaussian Elimination
• Get 0z in all other equations1x + 2y - 1z = 2 1 * R3 + R1 --> R10x + 1y - 6z = 0 6 * R3 + R2 --> R20x + 0y + 1z = 0.5
1x + 2y + 0z = 2.50x + 1y + 0z = 30x + 0y + 1z = 0.5
Gaussian Elimination
• Finish my incomplete step• Get 0y in all other equations1x + 2y + 0z = 2.5 -2 * R2 + R1 --> R10x + 1y + 0z = 30x + 0y + 1z = 0.5
1x + 0y + 0z = -3.50x + 1y + 0z = 30x + 0y + 1z = 0.5
Solve the system of equations
-3x − 9y = -6-3x − 13y = -8
a) x = -2, y = 0
b) x = 0, y = 8/13
c) x = 1/2, y = 1/2
d) x = -1/2, y = -1/2
e) None of the above
-3x − 9y = -6 (-1/3)*R1 ->R1
-3x − 13y = -8
1x + 3y = 2
-3x − 13y = -8 3R1 + R2 -> R2
1x + 3y = 2
0x − 4y = -2 (-1/4)R2 -> R2
1x + 3y = 2 (-3)R2 + R1 -> R1
0x + 1y = ½
1x + 0y = 1/2
0x + 1y = 1/2C
-3x − 9y = -6 (-1/3)*R1 ->R1
-3x − 13y = -8
1x + 3y = 2
-3x − 13y = -8 3R1 + R2 -> R2
1x + 3y = 2
0x − 4y = -2 (-1/4)R2 -> R2
1x + 3y = 2 (-3)R2 + R1 -> R1
0x + 1y = ½
1x + 0y = 1/2
0x + 1y = 1/2
What is the system of equations corresponding to the augmented matrix below?
3
4
2
3
1
2
a) 2x+3y = 4, x + 2y = 3
b) 3x+2y = 4, 2x + y = 3
c) 2x+y = 4, 3x + 2y = 3
d) x+y = 4, x + 2y = 3
e) None of the above
What is the system of equations corresponding to the augmented matrix below?
3
4
2
3
1
2
a) 2x+3y = 4, x + 2y = 3
Solving a system of equations on your calculator (and showing work)
• Solve4x + 8y - 4z = 82x + 3y + 4z = 45x + 8y + 1z = 7
In my calculator, I set the matrix [A]
Then I used the command rref([A])
The calculator output was
So the answer is
x=-3.5y=3z=0.5
Special situations
• If, at the end you wind up with something impossible, then there are NO SOLUTIONS
• Example:
The last row:0x + 0y = 1 is impossible,So there are NO SOLUTIONS.
Special situations
• If, at the end you wind up with something that is always true, then there are INFINITELY MANY SOLUTIONS
• Example:
The last row:0x + 0y = 0 is always true,So there are INFINITELY MANY SOLUTIONS.
Solve the following system.
2
1
52
z
zy
zyx
a) x = 0, y = 3, z = 2
b) x = 5, y = 3, z = 2
c) x = 1, y = 3, z = 2
d) x = -2, y = 3, z = 2
e) None of the above
2
1
52
z
zy
zyx
x=-2y=3z=2 D