Alg1131W10T2 - Deductions based on row-echelon form
Sunday, 13 May 2018
11:46 PM
Created with Microsoft OneNote 2016.
![Machine generated alternative text:
The point of row reduction is to simplify a system of equations
The system of equations in variables x , y z and w
has augmented matrix
x — 2y + w
1
—1
2
—1
1
o
1
2
1
1
The reduced row-echelon form of this is obtained by first eliminating entries below the leading entries of rows, from top to bottom, and then secondly eliminating entries above leading
entries of rows, from bottom to top
In a reduced row-echelon matrix [select all that apply]
41 every leading entry is 1
every entry below a leading entry is zero
every entry above a leading entry is zero
all fully zero rows are at the bottonm
Here is the reduced row-echelon form for A
This corresponds to the system of linear equations
1
o
o
o
1
o
2
1
o
3
o
0
1
—1
—1
3
2
Note that and correspond to completely equivalent systems of linear equations. From the reduced row-echelon form we can deduce that
there is a one-parameter family of solutions
O
O
there is a unique solution
there are no solutions.](Alg1131W10T2%20-%20Deductions%20based%20on%20row-echelon%20form_files/image001.png)
![Machine generated alternative text:
How did we deduce this from the row-echelon form U? We note that the leading entries are in the positions [select all that apply]
The leading columns of U are the columns containing these leading entries; namely [select all that apply]
column 1
column 2
column 3
column 4
column 5
—3=1
04-2=2
Since the last column (the augmented column) is not
O a leading column; we may conclude that the system has
O solutions.
To determine whether or not there is a unique solution or a several parameter family of solutions, we look at how many columns on the left hand side of U are not leading columns. This
number is
and so there are a
O parameter family of solutions.](Alg1131W10T2%20-%20Deductions%20based%20on%20row-echelon%20form_files/image002.png)
![Machine generated alternative text:
The 2017 US. presidential election had everyone talking. Could we have used what they said, and our prodigious skills in algebra, to predict the outcome?
People vote state-by-state for 538 electors; who then vote for the president The winner is the candidate what receives over 270 electoral votes.
Let D be the number of votes the Democrats' candidate receives, R be the number of votes the Republicans' candidate receives, and G be the number of votes the Green party's candidate
receives
Pundit 1 said 'The Democrats; the Republicans, and the Greens Party will receive a total of 500 electoral votes this year" _ As a linear equation this is
D+R+G=500
Note: your answer should be an equation like 3*R+4 *D+2
Pundit 2 said: 'The Green Party's candidate will never wim They'll only receive one quarter of the Democrat and Republican candidates votes combined't As a linear equation this is
Pundit 3 said "By themselves the Democrat and the Republican candidates will take 450 electoral votes this yeart As a linear equation this is
D+R=450
If we row-reduce these equations we find that [select all that apply]
we cannot predict a winner
politics is cheap entertainment.
the predictions of the political pundits is inconsistent](Alg1131W10T2%20-%20Deductions%20based%20on%20row-echelon%20form_files/image006.png)