IB Math AA HL Systems of Linear Equations — Complete Cheatsheet

Systems of Linear Equations is the gateway HL-only topic that links algebra, geometry of planes, and matrix methods. Sitting in AHL 1.16, it appears reliably on Paper 1 (full row-reduction shown) and Paper 2 (GDC rref()), and frequently as a Section B 8–12 mark question with a parametric twist. The topic itself is procedural, but the trap density is high: a single sign error in a multiplier wrecks the rest of the elimination, "no unique solution" is regularly confused with "no solution", and students forget to write the explicit parametric form when there are infinite solutions.

This cheatsheet condenses every method, scenario, and trap from AHL 1.16 into one page you can revise from. For the printable PDF, full notes, the tutorial booklet, and the marked-up solutions, scroll to the bottom for the download links and the gated full library.

§1 — Augmented Matrix & Row Operations AHL 1.16

From system to matrix

$$\begin{cases} a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3 \end{cases} \;\longrightarrow\; \left[\begin{array}{ccc|c} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \end{array}\right]$$

The bar separates coefficients (left) from constants (right). A missing variable on either side $\Rightarrow$ write 0, never blank.

Example setup

$3x - y + 2z = 7$;   $x + 4z = -1$ (no $y$: put 0);   $2x - 3y = 5$ (no $z$: put 0).

$$\left[\begin{array}{ccc|c} 3 & -1 & 2 & 7 \\ 1 & 0 & 4 & -1 \\ 2 & -3 & 0 & 5 \end{array}\right]$$

The three legal row operations

1. Swap: $R_i \leftrightarrow R_j$
2. Scale: $R_i \to k R_i$   ($k \neq 0$)
3. Replace: $R_i \to R_i + k R_j$

None of these change the solution set.

IB notation: $R_2 \to R_2 - 2R_1$ means the new $R_2$ equals the old $R_2$ minus $2 \times R_1$. Similarly $R_1 \to \tfrac{1}{3} R_1$ means divide row 1 by 3.

§2 — RREF Definition & Algorithm AHL 1.16

RREF — the four conditions

1. Leading entry (pivot) $= 1$.
2. The pivot is the only nonzero entry in its column.
3. Pivots move right going down.
4. Zero rows are at the bottom.

RREF algorithm — 5 steps

1. Write the augmented matrix.
2. Pivot (1,1): swap if needed; scale to get a 1; eliminate column 1 below.
3. Pivot (2,2): scale to 1; eliminate column 2 above and below.
4. Pivot (3,3): scale to 1; eliminate column 3 above and below.
5. Read the bottom row $\to$ classify the solution.

Bottom row diagnostic

Always inspect the bottom row first.

§3 — Scenario 1: Unique Solution AHL 1.16

Bottom row $[\,0\;0\;1\;|\;r\,]$. The full RREF is

$$\left[\begin{array}{ccc|c} 1 & 0 & 0 & p \\ 0 & 1 & 0 & q \\ 0 & 0 & 1 & r \end{array}\right]$$

Read off directly: $x = p,\; y = q,\; z = r$. IB term: consistent, independent. Geometry: three planes meet at one point.

Mini-example

$x + y + z = 6,\; 2x - y + z = 3,\; x + 2y - z = 2$. After RREF:

$$\left[\begin{array}{ccc|c} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array}\right] \;\Rightarrow\; (x, y, z) = (1, 2, 3).$$

Check: $1 + 2 + 3 = 6$ ✓.

§4 — Scenario 2: Infinite Solutions AHL 1.16

Bottom row $[\,0\;0\;0\;|\;0\,]$. The RREF looks like

$$\left[\begin{array}{ccc|c} 1 & 0 & a & p \\ 0 & 1 & b & q \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Let $z = t,\; t \in \mathbb{R}$ (free parameter). Back-substitute: $y = q - bt,\; x = p - at$. Write the general solution as

$$(x, y, z) = (p - at,\; q - bt,\; t).$$

IB term: consistent, dependent. Geometry: three planes share a line.

General solution as a line of intersection

$$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} p \\ q \\ 0 \end{pmatrix} + t \begin{pmatrix} -a \\ -b \\ 1 \end{pmatrix}, \quad t \in \mathbb{R}.$$

This links to AHL 3.14: this is the vector equation of the line of intersection of the planes.

Mini-example

After RREF of a $2 \times 3$ system:

$$\left[\begin{array}{ccc|c} 1 & 0 & \tfrac{1}{2} & \tfrac{17}{4} \\ 0 & 1 & -\tfrac{3}{2} & \tfrac{3}{4} \end{array}\right]$$

Let $z = t$: $x = \tfrac{17}{4} - \tfrac{1}{2} t,\; y = \tfrac{3}{4} + \tfrac{3}{2} t$.

§5 — Scenario 3: No Solution AHL 1.16

Bottom row $[\,0\;0\;0\;|\;k\,]$ with $k \neq 0$. The row reads $0 = k$ — a contradiction.

IB term: inconsistent. State: "The system has no solution" or "The system is inconsistent."

Geometric configurations

• The line of intersection of $\pi_1 \cap \pi_2$ is parallel to $\pi_3$.
• Two planes are parallel (and not coincident).
• The three planes form a triangular prism — pairwise intersections are three distinct parallel lines.

§6 — Parametric Systems AHL 1.16

The system contains an unknown $a$ or $b$ in its coefficients. Strategy:

1. Carry out RREF with $a, b$ as symbols.
2. Bottom row reduces to $[\,0\;0\;g(a)\;|\;h(b)\,]$.
3. "No unique solution" $\Rightarrow$ set $g(a) = 0$.
4. If $h(b) = 0$ $\Rightarrow$ infinite solutions; if $h(b) \neq 0$ $\Rightarrow$ no solution.

Classic IB structure: (a) Find $a$ for which there is no unique solution [4 marks]; (b) Find $b$ for consistency [1]; (c) State the general solution [3].

§7 — GDC: rref() on Paper 2 Paper 2

Enter the matrix as a $3 \times 4$ augmented matrix. Always include the constants column.

Paper 2: show the rref output in your working. Paper 1: show every row operation in IB notation — rref() alone scores zero on Paper 1.

§8 — Exam Attack Plan All sections

When you see this in the question — reach for that:

Eight big exam traps

1. A missing variable means write 0, not blank.
2. Sign error in the multiplier (negate then add).
3. Only the target row changes — never $R_1$ when targeting $R_2$.
4. "No unique" $\neq$ "no solution".
5. The general solution must be explicit with a parameter $t$.
6. GDC matrix must be $3 \times 4$, not $3 \times 3$ — include the constants column.
7. Verify by substituting back into all three equations.
8. Memorise the three geometric configurations for the no-solution case.

Worked Example — IB-Style Parametric System

Common Student Questions