Substitution Method Calculator
Solve two linear equations using substitution steps. Enter both equations, compute variable values, show determinants, checks, and explanatory equations clearly.
Equation Preview
Helping Notes
Write each equation in x and y. Fractions, negatives, and spaces are fine. 2x means 2·x.
We detect unique, infinite, or no solution using coefficients extracted from your equations and show clear checks.
Results
Parsed Standard Form
Determinants
Substitution Steps
Solution (x, y)
Check
What Is a Substitution Method Calculator?
A Substitution Method Calculator automates solving by replacement. For algebraic systems, one equation is rearranged to express a variable in terms of another, then substituted into the remaining equation(s) to find consistent solutions. For calculus, the same idea appears as u‑substitution: define a new variable so the integral simplifies from to . The tool shows each algebraic manipulation, maintains exact fractions, and renders responsive formulas so you can verify every step clearly across screens.
About the Substitution Method Calculator
In linear systems, substitution complements elimination and matrix methods. Given and , if then ; substitute into the first to solve for , then back‑substitute for . Nonlinear systems also benefit when one variable is naturally isolated (e.g., ). In integration, a judicious choice of absorbs a derivative factor , with limits adjusted in definite cases via . The calculator detects these patterns, handles piecewise domains, and presents both exact and numeric results with concise justifications.
Linear system (generic substitution):
Plug into first:
u‑substitution (indefinite):
u‑substitution (definite):
How to Use This Substitution Method Calculator
- Select Algebraic System or Integral mode.
- For systems: enter equations; choose a variable to isolate (e.g., solve the second for ).
- For integrals: enter the integrand; suggest or let the tool detect a suitable substitution and update limits if definite.
- Compute to see step‑by‑step substitution, simplification, and the final solution or antiderivative/value.
Examples
- Linear system: . From the second, . Substitute: .
- Nonlinear system: ⇒ , then .
- Indefinite integral: with ⇒ .
- Definite integral: , let ⇒ .
Formula Snippets Ready for Rendering
FAQs
When should I use substitution instead of elimination?
Use substitution when an equation is easy to isolate for one variable; elimination is convenient with aligned coefficients.
Does substitution work for nonlinear systems?
Yes. If one variable can be expressed as a function of the other, substitute and solve the resulting equation.
How do I choose a good u for an integral?
Look for an inner function whose derivative appears as a factor; typical patterns are composites, roots, and quotients.
What if the definite integral bounds aren’t converted?
Either change the bounds with or back‑substitute before evaluating from to .
Can substitution introduce extraneous solutions?
Yes, if you square or otherwise expand domain; always check solutions in the original system or integrand’s domain.
What if no simple substitution exists?
Try algebraic manipulation, trigonometric substitution, or numerical methods; the calculator will suggest alternatives when patterns fail.
Will the tool show exact fractions?
Yes. It keeps exact symbolic forms and also provides decimal approximations for quick checking.