Substitution Method Calculator — Solve Linear Systems Step by Step
Enter the coefficients of two linear equations (a₁x + b₁y = c₁ and a₂x + b₂y = c₂) and the calculator solves the system using the substitution method — showing every algebraic step and plotting the intersection of the two lines.
Express y from Equation 1: 2x + y = 7
Substitute y into Equation 2: x − y = 2
Solve for x
Back-substitute x = 3 into y = (7 − 2·x) / 1
Solution
- 1
Determinant det = a₁b₂ − a₂b₁
2 × -1 − 1 × 1 = -3 - 2
Numerator for x: c₁b₂ − c₂b₁
7 × -1 − 2 × 1 = -9 - 3
Numerator for y: a₁c₂ − a₂c₁
2 × 2 − 1 × 7 = -3 - 4
x = numerator ÷ det
-9 ÷ -3 = 3 - 5
y = numerator ÷ det
-3 ÷ -3 = 1
How does this calculator work?
Express one variable from Equation 1, substitute into Equation 2, and solve. The unique solution is x = (c₁b₂ − c₂b₁)/det and y = (a₁c₂ − a₂c₁)/det where det = a₁b₂ − a₂b₁. If det = 0 the lines are parallel or identical — no unique solution.
Formula
How this is calculated
The substitution method isolates one variable from one equation and replaces it in the other — turning two equations in two unknowns into a single equation in one unknown that you can solve directly. Concretely: from Equation 1 (a₁x + b₁y = c₁) express y (or x if the y-coefficient is zero), then substitute that expression into Equation 2. The resulting single-variable equation gives a numeric value; substituting that back into either original equation gives the second variable.
Geometrically, each equation is a straight line. The solution (x, y) is the point where the two lines cross. If the lines are parallel (same slope, different intercepts) the determinant a₁b₂ − a₂b₁ equals zero and there is no unique solution. If the lines are identical, infinitely many solutions exist — the calculator flags both cases.
The method works for any real coefficients, including fractions and negatives. Coefficients with a common factor do not need to be simplified before entering; the calculator handles the arithmetic exactly.
Frequently asked questions
Substitution is easiest when one equation already has a coefficient of 1 or −1 (e.g. x + 2y = 5), because isolating that variable produces no fractions. Elimination is often faster when both equations have large or awkward coefficients, because it avoids messy intermediate expressions.
The determinant (a₁b₂ − a₂b₁) is zero, meaning the two lines are either parallel (no solution) or identical (infinitely many solutions). Check whether the ratio a₁/a₂ equals b₁/b₂ — if c₁/c₂ also matches, the equations are the same line; if not, they are parallel.
This calculator handles exactly two equations and two unknowns. For 3×3 systems you would apply substitution or elimination repeatedly, or use matrix methods (Gaussian elimination). Larger systems are typically solved with software.
Also known as
TG we-Calculate Editorial Team. (2026). Substitution Method Calculator — Solve Linear Systems Step by Step [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/substitution-method-calculator
TG we-Calculate Editorial Team. "Substitution Method Calculator — Solve Linear Systems Step by Step." TG we-Calculate. 2026. https://we-calculate.com/calculator/substitution-method-calculator.
TG we-Calculate Editorial Team, "Substitution Method Calculator — Solve Linear Systems Step by Step," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/substitution-method-calculator
@misc{wecalculate_substitution_method_calculator, title = {Substitution Method Calculator — Solve Linear Systems Step by Step}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/substitution-method-calculator}}, year = {2026}, note = {TG we-Calculate} }
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