Systems and Augmented Matrices

Strip away the letters and keep the numbers.

12 problems · about 22 minutes · A-REI.C.6, A-REI.C.8

What this lesson teaches

The student writes a linear system as an augmented matrix and reads one back.

A linear system keeps every unknown to the first power and never multiplies unknowns together.
Stripping the letters leaves an augmented matrix: coefficients, a rule, then constants.
Three reversible row operations rearrange the matrix without changing the solutions.

Straightforward practice. Get the method working first.

5 problems

5x − 3y = 11. What is the coefficient of y?
Answer -3
Why −3.
What does the vertical rule in an augmented matrix mark?
Answer The split between the coefficients and the constants.
Why It separates A from b.
x + 5y = 7. What goes in the constant column?
Answer 7
Why The right-hand side.
x + 5y = 7. What is the coefficient of x?
Answer 1
Why An unwritten coefficient is 1.
2x − y = 0. What is the coefficient of y?
Answer -1
Why The minus sign belongs to it.

The same ideas with more to keep track of.

3 problems

A system of 2 equations in 2 unknowns. How many columns does the augmented matrix have?
Answer 3
Why Two plus one.
How many legal row operations are there?
Answer 3
Why Swap, scale, and add a multiple.
Multiplying a row by 0. Is that legal? 1 yes, 0 no.
Answer 0
Why It would destroy the equation and cannot be undone.

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

4 problems

Sort each move by whether it is a legal row operation.
Answer Legal: Swap two rows, Multiply a row by 5, Add twice row 1 to row 2 · Not allowed: Multiply a row by 0
Why A legal move must be reversible.
Do row operations change the solution set? 1 yes, 0 no.
Answer 0
Why That is exactly why they are allowed.
The Reading: The row [4, −2 | 10] stands for 4x − 2y = 10. What is the coefficient of y?
Answer -2
Why −2.
The Size: A system of 3 equations in 4 unknowns. How many columns does the augmented matrix have?
Answer 5
Why 5.