Solving a system of equations means finding values that satisfy all equations simultaneously. The three standard techniques each have a sweet spot — knowing which to pick saves time on every homework set.

Method 1: Substitution

Best when one variable is already isolated (or easy to isolate).

Procedure:

Solve one equation for one variable.
Substitute that expression into the other equation.
Solve the resulting one-variable equation.
Back-substitute to find the second variable.

Example: $\begin{cases} y = 2x + 1 \\ 3x + y = 11 \end{cases}$

$y$ is already isolated. Substitute into the second: $3x + (2x + 1) = 11$ , so $5x = 10$ , $x = 2$ .
Back-substitute: $y = 2(2) + 1 = 5$ .
Solution: $(2, 5)$ .

Method 2: Elimination (Linear Combination)

Best when coefficients line up to cancel a variable by adding / subtracting.

Procedure:

Multiply one or both equations by constants so a variable's coefficients are opposites (e.g. $+3y$ and $-3y$ ).
Add the equations to eliminate that variable.
Solve the remaining one-variable equation.
Back-substitute.

Example: $\begin{cases} 2x + 3y = 12 \\ 4x - 3y = 6 \end{cases}$

$3y$ and $-3y$ already opposite. Add: $6x = 18$ , $x = 3$ .
Back-substitute: $2(3) + 3y = 12$ , $3y = 6$ , $y = 2$ .
Solution: $(3, 2)$ .

Method 3: Matrix methods

For larger systems (3+ variables) or computer-aided solving:

Cramer's rule: $x_i = \det(A_i) / \det(A)$ where $A_i$ is $A$ with the $i$ -th column replaced by the constants. Works for any size, but $\det$ computation grows fast.
Gaussian elimination: row-reduce the augmented matrix $[A | \vec{b}]$ to row echelon form, back-substitute. The standard method for large systems.
Inverse matrix: $\vec{x} = A^{-1} \vec{b}$ . Works only if $A$ is square and invertible (non-zero determinant).

For 2×2 systems by hand, substitution or elimination almost always wins. Matrix methods shine for 3+ variables.

Three possibilities for the solution set

Every linear system has exactly one of:

One unique solution: lines (or planes) intersect at one point.
No solution: equations contradict (parallel lines that don't meet) — system is inconsistent.
Infinite solutions: equations describe the same line / plane — system is dependent.

Algebraic signal:

" $x = 5$ " → unique.
" $0 = 7$ " → contradiction → no solution.
" $0 = 0$ " → tautology → infinite solutions.

Common mistakes

Sign errors when distributing during substitution. Bracket carefully.
Forgetting to multiply both sides during elimination scaling.
Stopping after finding $x$ . Both variables matter; back-substitute.
Ignoring inconsistency. If you get $0 = 7$ , that's the answer ("no solution"), not a calculation error.

Try it yourself

Drop any system into our free System of Equations Solver — the AI picks substitution / elimination automatically and shows every step.

Three Ways to Solve Systems of Equations

Master systems of equations with substitution, elimination, and matrix methods. Worked examples for 2×2 and 3×3 systems, plus when each method shines.