System of Equations Calculator

Linear equations

Enter your equations

Put each coefficient beside its variable and the constant after the equals sign. The solver accepts signed whole numbers, decimals, zero, and scientific notation.

Result

Solution and classification

Read the solution first, then verify the ranks and reduced matrix.

One unique solutionx = 4; y = 2

x: 4
y: 2

Why this resultBoth ranks equal 2, so every variable has one determined value.

Final row-reduction state

Reduced augmented matrix (RREF)

The right side is separated from the variable columns.
Row	x	y	Right side
R1	1	0	4
R2	0	1	2

How it was solved

From equations to classification

1
Interpret the bounded list as row-major coefficients with the final entry of each row on the right side.2 rows × 3 augmented columns
2
Scale equations, choose the largest available pivot in each coefficient column, and eliminate above and below it.Gauss-Jordan elimination; relative zero threshold 1e-12
3
Compare coefficient and augmented ranks before returning any variable values.rank(A)=2; rank([A|b])=2; n=2

Technical details and assumptions

rank(A): 2
rank([A|b]): 2
Variables: 2
Zero threshold: 1e−12

This is a numeric 2×2/3×3 linear solver, not a computer algebra system. It does not parse equations, fractions as text, parameters, inequalities, powers, or nonlinear terms.
Classification uses row scaling and a 1e-12 pivot threshold. Nearly singular or badly conditioned decimal systems can be sensitive to small input changes; verify such work with exact arithmetic or a numerical-analysis tool.
Parenthetical fractions are bounded rational approximations of the computed decimal solution, not proof that the original decimal coefficients were exact fractions.

Production-authorized calculationFormula 1.0.0. Supports bounded real 2×2 and 3×3 linear systems.

Simultaneous-constraint solver

Solve a linear system by preserving every equation at once

A system solution is not a value that works in one convenient row. It is an ordered pair or triple that satisfies every original equation, with rank revealing whether the constraints meet once, coincide, or conflict.

Step-by-step guide · Simultaneous-constraint solver

Ticket counts create a two-equation model with an integer check

Available sizes: 2×2 and 3×3 linear systems
Unique case: one value for every variable
Other cases: inconsistent or dependent

Translate both facts
The count constraint is a+y=180. The money constraint is 14a+8y=2,040. The units differ between equations, but both involve the same unknown counts.
Eliminate one variable
Multiply the first equation by 8 to obtain 8a+8y=1,440. Subtract it from the revenue equation: 6a=600, so a=100. Then y=80.
Check both constraints and context
100+80=180 and 14(100)+8(80)=2,040. Both counts are nonnegative integers, so the algebraic result also fits the ticket context.

Solve a linear system by preserving every equation at once: A community event sold 180 tickets. Adult tickets cost $14, youth tickets cost $8, and total receipts were $2,040. Let a be adult tickets and y be youth tickets.

Each augmented row is one complete constraint

A row [a,b,c] in a two-variable system represents ax+by=c. A three-variable row has four entries. Keeping coefficients in fixed x, y, z, right-side order prevents a missing variable from shifting later entries; if a term is absent, enter a zero coefficient. Row operations may swap rows, multiply a row by a nonzero constant, or add a multiple of one row to another without changing the solution set.

A unique system has a pivot for every variable. A contradiction row such as [0,0|1] proves no solution. A zero row [0,0|0] can signal a redundant equation and free variables, producing infinitely many solutions. For a single equation in x, the solve equation calculator is clearer; one line cannot determine a generic pair of unknowns.

Ticket counts create a two-equation model with an integer check

A community event sold 180 tickets. Adult tickets cost $14, youth tickets cost $8, and total receipts were $2,040. Let a be adult tickets and y be youth tickets.

Translate both facts
The count constraint is a+y=180. The money constraint is 14a+8y=2,040. The units differ between equations, but both involve the same unknown counts.
Eliminate one variable
Multiply the first equation by 8 to obtain 8a+8y=1,440. Subtract it from the revenue equation: 6a=600, so a=100. Then y=80.
Check both constraints and context
100+80=180 and 14(100)+8(80)=2,040. Both counts are nonnegative integers, so the algebraic result also fits the ticket context.

Graphs and elimination expose complementary system evidence

Graphing shows intersection geometry

Two distinct lines can intersect once, remain parallel, or coincide. After substitution reduces a supported system to one variable, the algebra calculator can audit the resulting expression or equation without replacing the system classification.

Row reduction records algebraic classification

Reduced row echelon form exposes pivot columns and contradictions directly. It extends naturally from 2×2 to 3×3 systems and supports an auditable rank explanation.

Slope diagnoses two-line special cases

Equal slopes with different intercepts indicate parallel inconsistency; equal slopes and intercepts indicate dependence. The finding slope calculator can confirm a slope from two points but does not solve the simultaneous coefficient matrix.

System errors often occur before elimination begins

Define variables with units, then write one full equation for each independent fact in the story.
Keep the same variable order in every row and insert explicit zeros for missing terms.
Do not divide by a prospective pivot until checking whether it is zero; row swapping may preserve a valid route.
Substitute the final ordered pair or triple into every original equation. For a quadratic intersection problem, use the quadratic formula calculator after substitution creates a second-degree equation rather than forcing it into this linear solver.

Important limitations

System of Equations Calculator is limited to the finite inputs, modes, and outputs stated for the System of equations calculator method. Check system size and 2×2 augmented coefficient rows, validation messages, the equation, and the stated display precision before relying on system classification, solution, x, and y.

Review the stated scope, calculation method, assumptions, supported inputs, and effective dates, and obtain qualified advice before using a high-stakes result.

This formula passed its registered engineering tests and is published under an explicit site-owner policy authorization. No independent expert, QA, editorial, legal, or third-party review is claimed. Verify consequential results with a qualified professional.

Simultaneous-constraint solver

Solve a linear system by preserving every equation at once

Step-by-step guide · Simultaneous-constraint solver

Ticket counts create a two-equation model with an integer check

Available sizes: 2×2 and 3×3 linear systems
Unique case: one value for every variable
Other cases: inconsistent or dependent

Translate both facts
The count constraint is a+y=180. The money constraint is 14a+8y=2,040. The units differ between equations, but both involve the same unknown counts.
Eliminate one variable
Multiply the first equation by 8 to obtain 8a+8y=1,440. Subtract it from the revenue equation: 6a=600, so a=100. Then y=80.
Check both constraints and context
100+80=180 and 14(100)+8(80)=2,040. Both counts are nonnegative integers, so the algebraic result also fits the ticket context.

Each augmented row is one complete constraint

Ticket counts create a two-equation model with an integer check

A community event sold 180 tickets. Adult tickets cost $14, youth tickets cost $8, and total receipts were $2,040. Let a be adult tickets and y be youth tickets.

Translate both facts
The count constraint is a+y=180. The money constraint is 14a+8y=2,040. The units differ between equations, but both involve the same unknown counts.
Eliminate one variable
Multiply the first equation by 8 to obtain 8a+8y=1,440. Subtract it from the revenue equation: 6a=600, so a=100. Then y=80.
Check both constraints and context
100+80=180 and 14(100)+8(80)=2,040. Both counts are nonnegative integers, so the algebraic result also fits the ticket context.

Graphs and elimination expose complementary system evidence

Graphing shows intersection geometry

Row reduction records algebraic classification

Reduced row echelon form exposes pivot columns and contradictions directly. It extends naturally from 2×2 to 3×3 systems and supports an auditable rank explanation.

Slope diagnoses two-line special cases

System errors often occur before elimination begins

Define variables with units, then write one full equation for each independent fact in the story.
Keep the same variable order in every row and insert explicit zeros for missing terms.
Do not divide by a prospective pivot until checking whether it is zero; row swapping may preserve a valid route.
Substitute the final ordered pair or triple into every original equation. For a quadratic intersection problem, use the quadratic formula calculator after substitution creates a second-degree equation rather than forcing it into this linear solver.

System of Equations Calculator

Enter your equations

Fill in each equation

Solution and classification

Reduced augmented matrix (RREF)

From equations to classification

Solve a linear system by preserving every equation at once

Ticket counts create a two-equation model with an integer check

Translate both facts

Eliminate one variable

Check both constraints and context

Each augmented row is one complete constraint

Ticket counts create a two-equation model with an integer check

Graphs and elimination expose complementary system evidence

Graphing shows intersection geometry

Row reduction records algebraic classification

Slope diagnoses two-line special cases

System errors often occur before elimination begins

Important limitations

Enter your equations

Fill in each equation

Solution and classification

Reduced augmented matrix (RREF)

From equations to classification

Solve a linear system by preserving every equation at once

Ticket counts create a two-equation model with an integer check

Translate both facts

Eliminate one variable

Check both constraints and context

Each augmented row is one complete constraint

Ticket counts create a two-equation model with an integer check

Graphs and elimination expose complementary system evidence

Graphing shows intersection geometry

Row reduction records algebraic classification

Slope diagnoses two-line special cases

System errors often occur before elimination begins