Solution Set¶

Definition¶

The solution set of an equation or system is the set of all values of the unknowns that satisfy it.

Interpretation¶

The solution set records all values of the unknowns that work, not just one example. For a system of linear equations, a solution must satisfy every equation simultaneously, so the solution set is the intersection of the solution sets of the individual equations.

Three panels of two lines: crossing in one point, parallel with no common point, and coinciding along a whole line.

For two linear equations in two unknowns, each equation describes a line and the solution set is what the two lines have in common. If the lines cross, it is a single point. If they are parallel and distinct, it is empty, $S=\varnothing$ $\emptyset$ . If both equations describe the same line, it is that whole line, written with a parameter as in the Example $x=10-t$ $t$ , $y=t$ $t$ .

Here both equations describe one line, so the solution set is a whole line, written with a parameter $t$ . Change $\boldsymbol{b}_2$ to empty it, or a coefficient to shrink it to a point.

For linear systems over $\mathbb{R}$ or $\mathbb{C}$ , these three cases are the only possibilities, in any number of unknowns. If $\boldsymbol{x}_1\neq\boldsymbol{x}_2$ both solve $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ $b$ , then so does every $\boldsymbol{x}_1+t(\boldsymbol{x}_2-\boldsymbol{x}_1)$ $x_{1})$ , because $\boldsymbol{A}(\boldsymbol{x}_2-\boldsymbol{x}_1)=\boldsymbol{b}-\boldsymbol{b}=\boldsymbol{0}$ $0$ ; so a linear system never has exactly two solutions. Nonlinear equations behave differently, as $x^2=1$ with solution set $\{-1,1\}$ shows. More precisely, when there is a solution $\boldsymbol{x}_p$ , every solution is $\boldsymbol{x}_p$ plus a solution of $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$ $0$ , and the solutions of $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$ form a subspace. That subspace is later called the kernel, and the shifted copy an affine subspace.

A system without solutions still has a solution set, the empty set. Giving one solution when there are infinitely many does not describe it.

Key Results¶

Equivalent systems of equations have equal solution sets.
Over $\mathbb{R}$ or $\mathbb{C}$ , a consistent linear system (one with at least one solution) with free variables has infinitely many solutions, usually described using parameters.

Examples¶

The solution set of $x^2=1$ over $\mathbb{R}$ is $\{-1,1\}$ .
The solution set of $x+y=10$ can be written as $x=10-t$ $t$ , $y=t$ $t$ , with $t\in\mathbb{R}$ $R$ .

Prerequisites¶

System of Equations

Sources¶

Math II: Linear Algebra · Week 3

Aliases: set of solutions Keywords: solution space Symbols: complex numbers, element of, real numbers

Last updated: 20 October 2026 03:02