Teacher page: Systems of linear equations

The one thing to do with the animation

Ask what 0 = 0 means before you show the third case.

Most will say they have made a mistake. It means the second equation told them nothing new, so the solutions form a whole line. 0 = 5 means the equations contradict and nothing satisfies both.

Neither is an error. Students who believe every system has one solution hunt for a slip that is not there, lose the time, and then write nothing.

Why row operations are allowed

Swapping rows, scaling a row, and adding a multiple of one row to another all leave the solution set untouched. That is why the method works, and it is worth one minute rather than presenting the moves as a ritual.

And back-substitution is the only check left. Once the rows have been manipulated there is no original working to review, so substituting into every original equation is not optional.

The answers

1. x	Adding gives 3x = 9, so x = 3.
2. det[[1,1],[2,2]]	1(2) - 1(2) = 0.
3. Final row 0 = 5	B, no solutions; the equations contradict.

Where the marks go

1 markNaming which of the three outcomes, in words.

1 markAn infinite solution set described with a parameter.

1 markSubstituting back into every original equation.

What each wrong answer tells you

They give	What it means
1 (Q1)	Gave y.
4 (Q2)	Added instead of subtracting in the determinant.
D, an error (Q3)	The belief this page exists to remove. 0 = 5 is a result, not a slip.
C (Q3)	Confused 0 = 5 with 0 = 0. One is impossible, the other is always true.

Other things they will say

"Can I just use the calculator?" For a well-behaved system, yes. It will not tell you clearly which of the three cases you are in, and that is usually what the question is about.

"How do I write infinitely many?" With a parameter: x = t, then y and z in terms of t. Writing "infinitely many" alone leaves marks behind.

"What does a zero determinant mean?" Not exactly one solution. It does not distinguish none from infinitely many, so you still have to look at the rows.

A possible order

	What is happening
1	The three cases on screen. What does 0 = 0 mean?
2	Row operations, and why they are allowed.
3	A 3x3 worked fully, with back-substitution.
4	A system with no solution and one with infinitely many.
5	Determinant as a quick warning.

Two things not to say

Do not present every example as having one solution. It builds the exact expectation that wastes exam time.

Do not accept "infinitely many" without a parameterised answer.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

Solve three equationsSolve x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2.
x = 1, y = 2, z = 3.
Check itSubstitute back into all three.
6, 3 and 2. All three must be checked, not one.

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

Interpret geometricallyWhat does a unique solution to three equations in three unknowns mean geometrically?
Three planes meeting at a single point.
No solutionDescribe what the planes do when a system has no solution.
They have no common point: either two are parallel, or the three meet pairwise in parallel lines forming a triangular prism with nothing in the middle.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

Infinitely manyA system reduces to 0 = 0 in its last row. State what that means and how the answer is written.
The planes meet in a whole line, so there are infinitely many solutions. The answer is written parametrically, with one variable as the parameter, not as a single triple.
Find the awkward valueFor what value of k does x + y = 1 and 2x + ky = 2 fail to have a unique solution?
k = 2, where the second equation is twice the first, so the lines coincide and every point on the line is a solution. Any other k gives exactly one.

Practicalities

Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.