Simultaneous Linear Equations

Explore systems of linear equations. See how the solution corresponds to the point where two lines intersect. Discover the geometric configurations of unique, zero (parallel), and infinite (coincident) solutions.

Interactive Sandbox

Systems Graphing Plane

Adjust sliders to change the coefficients of Line 1 ($a_1x + b_1y = c_1$) and Line 2 ($a_2x + b_2y = c_2$). Observe the intersection update dynamically on the grid.

Graph: ax + by = c

Equations Display

Line 1 Coefficients

Coefficient a₁ 1.0
Coefficient b₁ 1.0
Constant c₁ 3.0

Line 2 Coefficients

Coefficient a₂ 1.0
Coefficient b₂ -1.0
Constant c₂ 1.0
Algebraic Solver

Step-by-Step Cramer's Determinants

Understand how Cramer's rule utilizes 2x2 matrix determinants to systematically solve two-variable simultaneous equations.

Cramer's Rule Resolution


System State Details


Unique Solution

Mathematical Description

Calculated Solutions

x = 2.00, y = 1.00

Evaluation

Conceptual Quiz

Test your understanding of linear systems, stability states, determinants, and solution methods.

Question 1 of 5 Score: 0/0

If the determinant D = a₁b₂ - a₂b₁ is equal to zero, and Dx ≠ 0, how many solutions exist?

Correct Answer!

Explanation text goes here...

Systems Cheat Sheet


  • Consistent system: At least one solution exists.
  • Inconsistent: Lines are parallel (No solution, D = 0).
  • Dependent: Lines are coincident (Infinite solutions, D = 0).
  • Cramer determinant rule: D = a₁b₂ - a₂b₁