What are Simultaneous Equations?
A system of simultaneous (linear) equations is a set of equations with the same variables that must all be true at the same time. The solution is the set of variable values satisfying every equation simultaneously.
a₂x + b₂y = c₂
Cramer's Rule: x = Dₓ/D, y = D_y/D
D = |a₁ b₁; a₂ b₂| = a₁b₂ − a₂b₁
Methods of Solving
| Method | Best for |
|---|---|
| Substitution | When one equation is easy to isolate |
| Elimination | When coefficients can be matched easily |
| Cramer's Rule | Systematic; works for any n×n system |
| Matrix (Row reduction) | Large systems; used by computers |
| Graphing | Visualizing approximate solutions |
Example (2×2)
2x + 3y = 8 x − y = 1
D = (2)(−1) − (1)(3) = −2 − 3 = −5
Dₓ = (8)(−1) − (1)(3) = −8 − 3 = −11 → x = −11/−5 = 2.2
D_y = (2)(1) − (1)(8) = 2 − 8 = −6 → y = −6/−5 = 1.2
Check: 2(2.2)+3(1.2)=4.4+3.6=8 ✓ 2.2−1.2=1 ✓
Frequently Asked Questions
What if the determinant D = 0?
If D = 0, Cramer's Rule fails — the system is either inconsistent (no solution: parallel lines) or dependent (infinite solutions: same line). The calculator will detect and report this case.
What is a consistent vs inconsistent system?
Consistent: the system has at least one solution. Inconsistent: no solution exists (the lines are parallel — same slope, different intercepts). Dependent: infinite solutions (both equations represent the same line).
Can this solve non-linear equations?
No — this calculator handles only linear equations (no x², xy, sin(x), etc.). For non-linear systems, numerical methods like Newton's method are required.