Gaussian Elimination

Clear out entries until the answer is obvious.

12 problems · about 24 minutes · A-REI.C.6, A-REI.C.8

What this lesson teaches

The student row reduces an augmented matrix to echelon form and back-substitutes.

Elimination clears entries below each pivot until the matrix is a staircase.
The bottom row then has one unknown, and back substitution unravels the rest.
Reduced row echelon form goes further and leaves the answers in the last column.

Straightforward practice. Get the method working first.

5 problems

A row reads [0, 5 | 20]. What is y?
Answer 4
Why 4.
What shape is Gaussian elimination aiming for?
Answer A staircase of zeros below the pivots.
Why Echelon form.
A row reads [0, 3 | 9]. What is y?
Answer 3
Why 3y = 9.
A row reads [0, −2 | 8]. What is y?
Answer -4
Why Divide by −2.
Row 1 is [1, 1 | 10] and y = 4. What is x?
Answer 6
Why x + 4 = 10.

The same ideas with more to keep track of.

3 problems

Clearing the entry 5 below a pivot of 1. What multiple of the pivot row is subtracted?
Answer 5
Why Whatever makes it zero.
In reduced row echelon form, what is every pivot scaled to?
Answer 1
Why A leading 1.
A 3 by 3 system fully reduced. How many pivots does it have if it has a unique solution?
Answer 3
Why One per unknown.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Order the steps of Gaussian elimination.
Answer 1. Choose a pivot in the first column 2. Clear every entry beneath that pivot 3. Move to the next column and repeat 4. Back-substitute from the bottom row upward
Why Substitution comes only after the staircase is built.
A row reads [0, 1 | −6]. What is y?
Answer -6
Why Read it directly.
The Bottom Row: A reduced row reads [0, 4 | 12]. What is y?
Answer 3
Why 3.
The Substitution: Row 1 reads [1, 2 | 7] and you know y = 2. What is x?
Answer 3
Why 3.