Linear Equations Practice Arena — step solver, speed drill, and cheat sheet.
Interactive step-by-step solver
Enter coefficients for both equations, pick a method, and solve step by step.
Equation 1 a₁x + b₁y = c₁
x +
y =
Equation 2 a₂x + b₂y = c₂
x +
y =
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Quick-reference cheat sheet
Definitions
Consistent pairHas at least one solution — lines intersect or coincide
Inconsistent pairNo solution — lines are parallel and never meet
Dependent pairInfinite solutions — both equations describe the same line
Ratio test — graphical method
a₁/a₂ ≠ b₁/b₂Intersecting lines — unique solution (consistent)
a₁/a₂ = b₁/b₂ = c₁/c₂Coincident lines — infinite solutions (dependent)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂Parallel lines — no solution (inconsistent)
Substitution method
Step 1 — IsolateExpress one variable from either equation (prefer coefficient = 1)
Step 2 — SubstitutePlug the expression into the other equation
Step 3 — Back-subSolve, then substitute back to find the second variable
True statement (18=18)Infinitely many solutions — dependent pair
False statement (0=−4)No solution — inconsistent pair
Elimination method
Step 1 — MatchMultiply equations to make one variable's coefficients equal
Step 2 — EliminateAdd (opposite signs) or subtract (same signs) to cancel
Step 3 — SolveSolve the single-variable equation that remains
Step 4 — VerifySubstitute back into both original equations to check
Which method to choose?
GraphicalVisualise only — unreliable for non-integer answers
SubstitutionBest when one variable has coefficient 1 or −1
EliminationBest when both equations have larger or matching coefficients
Word problem translation formulas
Two-digit number10x + y (tens=x, units=y) Reversed: 10y + x
Age problemWrite equations for two different time points, then substitute
Fraction problemCross-multiply each condition to clear denominators first
Cost / rateTotal = unit cost × quantity — form one equation per condition
