Topic 1.8 · teacher page · AI, SL and HL

A solution is where they cross

What the solver is finding, and the case it cannot find because it is not there.

The one thing to do with the animation

Tilt the second line until it is parallel, and ask what the calculator will say.

Students who have only ever typed into a solver have no picture at all, so a system with no solution is just an error message. Here it is obviously two lines that never meet.

The guide promises a unique solution in examinations. That is a promise about the questions, not about mathematics, and the distinction is worth making explicitly.

Substituting back is the only check available

When the method is "I typed it in", there is no working to review. Substituting both values into both equations takes ten seconds and catches a mistyped coefficient, which is the realistic failure here.

And graph a polynomial before trusting the roots. The graph says how many real roots to expect, so you know whether the solver gave you all of them.

The answers

1. x from the system3, with y = 1.
2. Largest root of the cubic3; the roots are 1, 2 and 3.
3. Parallel distinct linesB, no solutions.

Where the marks go

1 markEquations written in a consistent form before entry.

1 markAll roots given when the question says solve.

1 markA check by substitution, and an interpretation in context.

What each wrong answer tells you

They giveWhat it means
1 (Q1)Gave y.
9 (Q1)Gave 3x without dividing.
1 (Q2)Gave the smallest root; the question asked for the largest.
6 (Q2)Gave the sum of the roots, which is also their product here. Substituting would have caught it.
C, infinitely many (Q3)Confused parallel with identical. Identical lines give infinitely many; parallel distinct give none.

Other things they will say

"Why learn this if the calculator does it?" Because the calculator will answer a question you did not ask, and only the picture tells you whether the answer is sensible.

"What if there is no solution?" Say so, and say why: the lines are parallel, or the planes have no common point. That is a complete answer, not a failure.

"Roots, zeros or x-intercepts?" The same thing in three vocabularies. Questions switch between them deliberately.

A possible order

 What is happening
1Tilt to parallel. What will the calculator say?
22x2 systems, with substitution back every time.
33x3 systems and what three planes meeting means.
4Polynomials: graph first, then solve, then count.
5A context problem where the system comes from a real situation.

Two things not to say

Do not skip the picture because the solver is faster. The picture is the only part that survives into Topic 2.

Do not let "the calculator said so" be a method. It is not one, and it earns nothing when the entry was wrong.

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.

  1. 2x2 systemSolve 3x + 2y = 16 and x − y = 2.
    x = 4, y = 2
  2. Polynomial rootsSolve x³ − 7x + 6 = 0.
    x = 1, 2, −3

2In context

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

  1. Set up from contextThree coffees and two cakes cost 395 baht. One coffee and four cakes cost 365 baht. Find each price.
    Coffee 85 baht, cake 70 baht
  2. Interpret rootsA ball's height is h = 20t − 5t². Solve h = 0 and say what each root means.
    t = 0 and t = 4. The start, and landing 4 seconds later.

3Challenge

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

  1. Recognise no solutionSolve 2x + 3y = 7 and 4x + 6y = 15. What happens, and what does it mean?
    The calculator reports no solution. The lines are parallel: doubling the first gives 4x + 6y = 14, which contradicts 15.
  2. Count the roots firstSketch y = x³ − 3x + 1 roughly, say how many real roots to expect, then solve. Why does the sketch matter?
    Three real roots (about −1.88, 0.35, 1.53). The sketch tells you how many to look for, so you know whether the solver has given you all of them.

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.