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DIY Cost Tools

DIY Cost Tools

Linear System Calculator

Solve two linear equations in two unknowns when a unique solution exists. Arrange each equation in the displayed form before entering coefficients. Enter an explicit zero for any missing variable and include the sign of each coefficient. Once a solution appears, substitute x and y into both original equations. That independent check can reveal a transcription error even when the calculator correctly solves the coefficients you typed.

Enter every field, including explicit zeros. Decimal and scientific notation are accepted; commas, fractions and blank fields are rejected. Each input must be finite and no larger than 1 billion in absolute value.

Solution x3
Solution y2

How to use this calculator

  1. Match each input to its labeled coordinate, coefficient or matrix position.
  2. Check the signs and use one consistent coordinate convention.
  3. Select Calculate, or press Enter, and read the labeled results.

Formula and method

D = ad − bc; x = (ed − bf)/D; y = (af − ec)/D.

The calculation runs locally in your browser. Changing an input requires selecting Calculate again. If an input or mathematical condition is invalid, the previous result is hidden instead of leaving a stale answer visible.

Worked example

The equations x + y = 5 and x − y = 1 give x = 3 and y = 2. Substitution confirms 3+2=5 and 3−2=1.

Limitations and interpretation

This tool returns only numerically stable unique solutions. It does not distinguish infinitely many solutions from an inconsistent system when the determinant vanishes. Near-singular systems are rejected at the same relative 10⁻¹² threshold used by the inverse tool. Equations must be linear in x and y.

Results are displayed to approximately 10 significant digits. The underlying calculation uses ordinary floating-point numbers and is not exact symbolic arithmetic. Rounded results can differ slightly from hand calculations. For a measured quantity, choose precision that reflects the measurements rather than all the digits shown. Verify consequential calculations independently.

Frequently asked questions

What should I enter for a missing term?

Enter zero for its coefficient. Do not leave the field blank; blanks are treated as missing input.

Can I enter negative or decimal values?

Yes, provided the values meet the mathematical conditions described above. Type a normal hyphen for a negative sign. Do not type units, commas or a fraction such as 1/2; enter 0.5 instead.

Why did the calculator hide my previous result?

A missing or invalid input, undefined operation or unstable numerical case produces an explanation. Correct the input and calculate again to obtain a result for the current values.

Review notes and reference

Reviewed October 3, 2026. The stated default example and a boundary or algebraic check were tested against expected values. Input validation checks included blank fields, nonnumeric text and values outside the supported range. Wolfram MathWorld formula reference provides mathematical background. This tool is for calculation and education; its output alone does not validate a professional design.

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