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Solving Systems of Equations:
A brief tutorial for 15.053
•
This is a very brief introduction to solving systems of equations.
You
probably learned these techniques in high school, and you certainly
learned them if you had 18.06 or any other type of linear algebra
course.
Cleaver, an
MIT
Beaver
It doesn’t hurt to review.
And
perhaps you’ll learn something new.

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2
Elementary Row Operations
•
The following is a 3 x 4 matrix on which we will perform three types
of elementary row operations.
[1]
[2]
[3]
[4]
Row 1
1
2
3
4
Row 2
1
3
5
6
Row 3
0
1
2
3
The first elementary row operation consists of multiplying one
row by a constant.
We illustrate this by multiplying Row 2 by 3.
[1]
[2]
[3]
[4]
Row 1
1
2
3
4
Row 2
3
9
15
18
Row 3
0
1
2
3

3
The Second Elementary Row Operation
The second elementary row operation consists of adding a multiple of one
row of a matrix to another row of the matrix.
[1]
[2]
[3]
[4]
Row 1
1
2
3
4
Row 2
3
9
15
18
Row 3
0
1
2
3
We illustrate this by adding four times Row 1 to Row 3.
[1]
[2]
[3]
[4]
Row 1
1
2
3
4
Row 2
3
9
15
18
Row 3
4
9
14
19

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4
The Third Elementary Row Operation
The third elementary row operation consists of interchanging two rows of
a matrix.
[1]
[2]
[3]
[4]
Row 1
1
2
3
4
Row 2
3
9
15
18
Row 3
4
9
14
19
We illustrate this by interchanging Row 1 and Row 3.
[1]
[2]
[3]
[4]
Row 1
4
9
14
19
Row 2
3
9
15
18
Row 3
1
2
3
4