Equations in Three Unknowns

Three unknowns is where systematic elimination earns its keep. With two unknowns you can usually solve by inspection; with three you cannot. The right idea is to reduce the problem — use one equation to knock a variable out of the other two, leaving a 2x2 system you already know how to solve, then back-substitute. This is the engine that becomes Gaussian elimination in Chapter 17.

The standard shape

A system of three linear equations in three unknowns $x, y, z$ looks like:

Geometrically, each equation is a plane in 3D space. Three planes can meet at a single point, along a line, at no common point, or coincide. The four cases collapse to three for our purposes: unique solution, infinitely many, none.

The reduction strategy

1. Use one equation to eliminate $x$ from the other two (multiply and subtract).

2. You now have two equations in just $y$ and $z$ — a 2x2 system.

3. Solve that 2x2 system using elimination or substitution (§2.1).

4. Back-substitute into one of the original equations to recover $x$.

The widget walks the algorithm

(Type the system $x + y + z = 6$, $x - y = 2$, $y + z = 3$ into the matrix. Watch the elimination steps appear in real time. The determinant tells you whether the planes meet at a single point.)

How the determinant classifies the system

The coefficient matrix $A$ has a number called the determinant (defined formally in §17.4). For now, treat it as a black-box invariant of the system:

det(A) ≠ 0: unique solution. The three planes meet at exactly one point.

det(A) = 0, system consistent: infinitely many solutions. The three planes share a line or even coincide; one free parameter survives.

det(A) = 0, system inconsistent: no solution. The planes form a triangular prism or include a pair of parallel planes; there is no common point.

The widget's "classify" line implements exactly this trichotomy.

Try it

• Before entering anything: substitute $(x, y, z) = (1, 2, 3)$ into all three equations of $2x + y - z = 1$, $x - y + z = 2$, $3x + y = 5$. Then enter the system in the widget and verify the solution.
• Make the third row equal the sum of the first two. What happens to the determinant? To the solution set?
• Construct a system with infinitely many solutions by writing three equations whose left-hand sides are proportional but right-hand sides are also proportional.
• Construct an inconsistent system by keeping the same left-hand sides but breaking the proportionality on the right-hand sides.

A trap to watch for

When eliminating, students often multiply only the left side of an equation by some factor and forget the right side. Both sides scale together, always. If you multiply equation 1 by 2, the equation becomes $2a_1 x + 2b_1 y + 2c_1 z = 2d_1$. Forgetting to scale $d_1$ destroys the system silently — you get an answer, but it is wrong. The fix: when you write out the modified equation, treat the whole equation as one object and apply the operation to both sides simultaneously.

What you now know

You can solve any 3x3 linear system by reducing it to a 2x2 system, and you can use the determinant (later: any square system) to predict whether the system has a unique, multiple, or no solution. Chapter 17 will formalise this as Gaussian elimination on the coefficient matrix and introduce Cramer's rule for solving via determinants directly. Chapter 3 will switch gears and look at the underlying number system: the real numbers $\mathbb{R}$.

Quick check

References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 2, §2 — three equations in three unknowns and the elimination algorithm.
• Strang, G. (2016). Introduction to Linear Algebra, 5th ed. Wellesley-Cambridge. Chapter 2 — Gaussian elimination for general n x n systems with full pivoting rules.
• Axler, S. (2015). Linear Algebra Done Right, 3rd ed. Springer. Chapter 1 — the abstract view of linear systems as solutions of linear maps.