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Solve Simultaneous Equations (2x2) Calculator

Solve two linear equations with Cramer's rule.

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Solution x

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Overview

A system of two linear equations involves two algebraic statements that must be satisfied by the same pair of variables. Solving these equations finds the unique intersection point of two lines in a two-dimensional coordinate system.

Symbols

Variables

a = Coefficient a, b = Coefficient b, c = Coefficient c, d = Coefficient d, e = RHS e

Coefficient a
Variable
Coefficient b
Variable
Coefficient c
Variable
Coefficient d
Variable
RHS e
Variable
RHS f
Variable
Solution x
Variable
Solution y
Variable

Apply it well

When To Use

When to use: Use this when you have two distinct constraints affecting two unknown quantities. It is the standard method for finding the equilibrium of supply and demand or calculating mixtures in chemistry.

Why it matters: This form of modeling is the foundation of linear algebra and operations research. It allows professionals to optimize resource allocation, solve electrical circuit currents using Kirchhoff’s laws, and determine relative prices in economic models.

Avoid these traps

Common Mistakes

  • Swapping coefficients in the numerator.
  • Forgetting the denominator.

One free problem

Practice Problem

Practice Problem 1

Given the system 2x + 1y = 5 and 3x - 1y = 5, solve for the value of x.

Coefficient a2
Coefficient b1
RHS e5
Coefficient c3
Coefficient d-1
RHS f5

Solve for:

Hint: Try adding the two equations together to eliminate the y variable.

Practice Problem 2

A customer buys items where the first purchase of 1 unit of x and 2 units of y costs 10 dollars, and the second purchase of 2 units of x and 1 unit of y costs 11 dollars. Solve for the price of y.

Coefficient a1
Coefficient b2
RHS e10
Coefficient c2
Coefficient d1
RHS f11

Solve for:

Hint: Express x in terms of y using the first equation, then substitute it into the second.

Practice Problem 3

Solve the system 5x + 2y = 24 and 1x + 4y = 30 to find the value of x.

Coefficient a5
Coefficient b2
RHS e24
Coefficient c1
Coefficient d4
RHS f30

Solve for:

Hint: Multiply the first equation by 2 so that the y coefficients match.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Simultaneous linear equations
  2. Britannica: Simultaneous equations
  3. Halliday, Resnick, Walker, Fundamentals of Physics, 10th ed.
  4. Atkins' Physical Chemistry, 11th ed.
  5. Wikipedia: Dimensional analysis
  6. Wikipedia: System of linear equations
  7. Lay, David C. Linear Algebra and Its Applications. Pearson.
  8. Anton, Howard. Elementary Linear Algebra. Wiley.