LaTeX
x=\frac{d_1b_2-b_1d_2}{a_1b_2-b_1a_2},\quad y=\frac{a_1d_2-d_1a_2}{a_1b_2-b_1a_2}Variables
- a₁x + b₁y = d₁: first equation
- a₂x + b₂y = d₂: second equation
- a₁b₂ − b₁a₂: coefficient determinant
How to use this formula
Solves a nonsingular two-equation linear system using determinant ratios.
Important notes
- The denominator must be nonzero.
- Elimination is often more efficient for large systems.
Quick example
For x + y = 5 and x − y = 1, Cramer's rule gives x = 3 and y = 2.
Applicability, worked calculation, and verification
Assumptions and domain checks
- The denominator must be nonzero.
- For the Cramer's Rule for a 2×2 System, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold.
Worked example
For x + y = 5 and x − y = 1, Cramer's rule gives x = 3 and y = 2.
Common mistakes
- When copying Cramer's Rule for a 2×2 System, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established.
Continue the workflow
Use Cramer's Rule for a 2×2 System in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
x=\frac{d_1b_2-b_1d_2}{a_1b_2-b_1a_2},\quad y=\frac{a_1d_2-d_1a_2}{a_1b_2-b_1a_2}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Cramer's Rule for a 2×2 System used for?
Solves a nonsingular two-equation linear system using determinant ratios.
Can I copy this formula as LaTeX?
Yes. Copy x=\frac{d_1b_2-b_1d_2}{a_1b_2-b_1a_2},\quad y=\frac{a_1d_2-d_1a_2}{a_1b_2-b_1a_2} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.