System of Equations Calculator

Solves systems of linear equations using elimination/substitution method:

\[ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} \]

Equation 1 (x coefficient):

Equation 1 (y coefficient):

Equation 1 (constant):

Equation 2 (x coefficient):

Equation 2 (y coefficient):

Equation 2 (constant):

Solution:

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1. What is a System of Equations?

A system of equations is a set of two or more equations with the same variables. This calculator solves systems of two linear equations with two variables (x and y) using the elimination/substitution method.

2. How Does the Calculator Work?

The calculator uses the following method:

\[ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} \]

Where:

\( a_1, a_2 \) — Coefficients of x
\( b_1, b_2 \) — Coefficients of y
\( c_1, c_2 \) — Constants

Method: The calculator first checks the determinant of the coefficient matrix. If non-zero, it solves for x and y. If zero, it checks for consistency.

3. Types of Solutions

Unique Solution: When the determinant is non-zero, the system has exactly one solution.
No Solution: When the equations represent parallel lines (inconsistent system).
Infinite Solutions: When the equations represent the same line (dependent system).

4. Using the Calculator

Tips: Enter all six coefficients (a1, b1, c1, a2, b2, c2) for the two equations. The calculator will determine the solution type and provide the solution if it exists.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between substitution and elimination?
A: Substitution solves one equation for a variable and substitutes into the other. Elimination adds/subtracts equations to eliminate a variable.

Q2: Can this calculator handle non-linear equations?
A: No, this calculator is designed specifically for linear equations of the form ax + by = c.

Q3: What if I get "infinite solutions"?
A: This means the two equations are equivalent and all points on their common line are solutions.

Q4: Can I solve systems with more than two variables?
A: This calculator handles only two-variable systems. For more variables, you'd need more equations.

Q5: What does a zero determinant mean?
A: A zero determinant means the equations are either dependent (infinite solutions) or inconsistent (no solution).