2×2 Linear System by Substitution Calculator

Enter coefficients, select variable to isolate, and compute instantly with guided steps. See symbolic rearrangement and substitution performed with fraction handling and decimal options. Detect unique solutions, parallel lines, or coincident equations automatically too. Detailed step log clarifies algebra, including validation and residual checks for confidence. Export tables to CSV and PDF for reports.

Input

Substitution method
a₁
b₁
c₁
a₂
b₂
c₂

Results

Enter coefficients above and click “Solve by Substitution”.

Example Data

Click Load to populate the form quickly.

#a₁b₁c₁a₂b₂c₂IsolateFrom
123131-11y1
2124248x2
3112225y2

Formula Used

The substitution approach isolates one variable from a chosen equation, then replaces it in the other equation.

  • If isolating y from equation k: y = (Ck − Akx)/Bk. Substitute into other equation j to get:
    (Aj·Bk − Bj·Ak)·x = Cj·Bk − Bj·Ck, hence x = (Cj·Bk − Bj·Ck)/(Aj·Bk − Bj·Ak).
  • If isolating x from equation k: x = (Ck − Bky)/Ak. Substitute into other equation j to get:
    (Bj·Ak − Aj·Bk)·y = Cj·Ak − Aj·Ck, hence y = (Cj·Ak − Aj·Ck)/(Bj·Ak − Aj·Bk).
  • Determinant-like quantity D = A₁B₂ − A₂B₁ helps classify: D ≠ 0 unique solution; D = 0 either parallel inconsistent or coincident infinite solutions.

How to Use

  1. Enter a₁, b₁, c₁, a₂, b₂, c₂ for both equations.
  2. Select which variable to isolate and which equation to rearrange.
  3. Click Solve by Substitution to compute steps and results.
  4. Review classification, numeric solution, fraction form, and residual checks.
  5. Export the tables and steps area to CSV or PDF if needed.

FAQs

Substitution is convenient when one coefficient equals 1 or −1, making isolation simple. It also demonstrates algebraic relationships step-by-step, aiding understanding and symbolic manipulation clarity.

It examines D = a₁b₂ − a₂b₁ and proportionality among coefficients. If D ≠ 0, a unique solution exists; if D = 0, it distinguishes between parallel inconsistent and coincident infinite solutions.

Yes. You can pick x or y and the equation to rearrange. If the chosen coefficient is zero, the tool automatically switches to a sensible alternative.

Fractions can express exact values when coefficients are integers, while decimals help with numerical interpretation. Showing both provides exactness and readability simultaneously.

They verify the solution numerically by substituting x and y back into both equations. Values near zero confirm internal consistency and numeric stability.

About this calculator

Designed for clarity and auditability. Includes symbolic steps, fraction output, and classification of the system.

  • Automatic handling of degenerate isolation choices.
  • Residual checks to validate computed solution.
  • CSV and PDF exports for documentation and sharing.

Tip

For faster isolation, choose an equation where the target coefficient equals 1 or −1.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.