System of 2 Linear Equations Solver
Solve simultaneous 2-variable linear equations ($a x + b y = e$ and $c x + d y = f$) using Cramer's rule with step-by-step arithmetic.
x = 2
y = 3
Step-by-Step Cramer's Determinants Solution:
What is a System of 2 Linear Equations?
A System of Linear Equations consists of two or more algebraic equations sharing identical variables. For a standard 2-variable system:
1) $a x + b y = e$
2) $c x + d y = f$
The solution corresponds to the single geometric $(x, y)$ coordinate point where the two lines intersect on a Cartesian plane. If the lines are parallel, no solution exists; if the lines lie directly on top of each other, infinite solutions exist.
Key Capabilities
- Cramer's Rule Determinants: Computes $D$, $D_x$, and $D_y$ with step-by-step mathematical proofs.
- Inconsistent & Dependent System Detection: Accurately flags parallel lines (No Solution) and collinear lines (Infinite Solutions).
- Fraction and Decimal Support: Easily accommodates positive, negative, and fractional coefficient values.
- Instant Solution Verification: Automatically checks solutions back into original equations.
How to Solve Linear Equation Systems
- Enter the coefficients $a$, $b$, and constant $e$ for the first equation.
- Enter the coefficients $c$, $d$, and constant $f$ for the second equation.
- Click Solve System to view exact values for $x$ and $y$.
Frequently Asked Questions
How does Cramer's Rule work?
Cramer's Rule evaluates the determinant of the coefficient matrix $D = ad - bc$. Variable $x$ is found by replacing column 1 with constants ($D_x = ed - bf$) and dividing by $D$ ($x = D_x / D$). Similarly, $y = D_y / D$.
What does 'No Solution' mean geometrically?
No Solution indicates that the two equations describe distinct parallel lines with equal slopes ($m_1 = m_2$) that will never intersect anywhere on the coordinate plane.
Can this tool solve non-linear equations?
This specific solver is dedicated strictly to degree-1 linear systems. For quadratic polynomials ($x^2$), use our Quadratic Equation Solver.
Is this tool free?
Yes, it is 100% free with unlimited usage.