Starlight Tools

System of Equations Solver — Substitution, Elimination and Graphing

Enter two equations in standard form. Exact arithmetic and graphing run locally in your browser, and your values are not sent anywhere.

Get the exact ordered pair, decimals, determinant, checks in both equations, and worked substitution, elimination, and graphing views.

Enter the two equations

Equation 1: ax + by = c
Equation 2: dx + ey = f
2x + y = 7
x − y = 2

Use integers, finite decimals, or fractions such as -3/4. Enter 0 for a missing term.

Try an example

Solution

Solution set
Enter values to begin.
Decimal coordinates
—
System type—
Determinant—
Equation 1 check—
Equation 2 check—

Solution methods

Substitution steps will appear here.

Three ways to solve simultaneous equations

For the system ax + by = c and dx + ey = f, the determinant reveals whether there is one intersection:

Δ = ae − bd    |    x = (ce − bf) / Δ    |    y = (af − cd) / Δ

Substitution

Isolate one variable in either equation. Replace that variable in the other equation, solve the resulting one-variable equation, then substitute back.

Elimination

Scale the equations until one variable has matching coefficients. Subtracting the equations removes it and leaves a one-variable equation.

Graphing

Every nondegenerate linear equation is a line. The ordered pair where the two lines cross satisfies both equations.

Number of solutions

If Δ ≠ 0, there is one solution. If Δ = 0, consistency checks distinguish the same line from distinct parallel lines.

Worked example

For 2x + y = 7 and x − y = 2, isolate y in the first equation: y = 7 − 2x. Substitute it into the second: x − (7 − 2x) = 2, so 3x = 9 and x = 3. Then y = 7 − 2(3) = 1. The solution is (3, 1), and both lines meet at that point.

Calculation method, privacy, and limits

Integers, fractions, and finite decimals are converted to reduced fractions and processed with BigInt arithmetic. Classification, exact answers, and verification do not rely on floating-point rounding. Decimal answers are shown to up to 12 significant digits.

Each numeric part is limited to 30 digits to keep browser calculations responsive. The graph uses decimal coordinates for display and may be unavailable when values exceed the browser’s finite plotting range; the exact algebra result remains valid. Scientific notation and repeating-decimal notation are not accepted—enter an exact fraction instead. Inputs are not transmitted or stored. Last reviewed: 4 August 2026.

System of equations FAQs

How do you solve a system by substitution?

Isolate x or y in one equation, substitute that expression into the other equation, solve, and substitute back to find the remaining coordinate.

How do you solve a system by elimination?

Multiply one or both equations so a variable has matching or opposite coefficients, then add or subtract the equations to eliminate that variable.

How does graphing show the number of solutions?

One intersection means one solution. Distinct parallel lines mean no solution. Coincident lines are the same line and have infinitely many solutions.

What does the determinant tell me?

For this 2×2 system, Δ=ae−bd. A nonzero determinant guarantees one solution. A zero determinant requires a consistency check.

Can I enter fractions and decimals?

Yes. Every field accepts integers, finite decimals, or fractions such as -3/4. Finite decimals are converted to exact fractions.

Is this solver private?

Yes. Solving and graphing happen locally in your browser; the tool does not send or store your coefficients.

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