Mathematics ยท Algebra
Systems of Linear Equations: Elimination and Substitution
Reference entry · last updated September 20, 2026
A system of linear equations is a set of two or more linear equations in the same variables. A solution is an ordered assignment of values that satisfies every equation simultaneously. For two equations in two variables, the solution is the point where the two lines intersect.
1. First principles and equivalence
Two systems are equivalent when they have exactly the same solution set. Three operations preserve equivalence [1, 2]:
- Swap two equations.
- Multiply an equation by a nonzero constant.
- Add a multiple of one equation to another.
Both substitution and elimination are applications of these operations. Substitution uses equality to replace a variable; elimination adds scaled equations to cancel a variable.
2. Substitution method
Solve one equation for one variable, substitute that expression into the other equation, solve the resulting single-variable equation, then back-substitute.
Example. Solve \( y = 2x + 1 \) and \( x + y = 7 \). Substitute the first into the second:
Back-substitute: \( y = 2(2) + 1 = 5 \). The solution is \((2, 5)\). Substitution is efficient when one variable is already isolated or has coefficient \(1\).
3. Elimination method
Multiply one or both equations by constants so that the coefficients of one variable become opposites, then add the equations to cancel it.
Example. Solve \( 3x + 2y = 16 \) and \( 2x - 2y = 4 \). The \(y\) coefficients are already opposites, so add directly:
Back-substitute: \( 3(4) + 2y = 16 \Rightarrow 2y = 4 \Rightarrow y = 2 \). The solution is \((4, 2)\). Elimination is efficient when coefficients align or are easily matched.
4. Choosing a method
| Situation | Preferred method |
|---|---|
| A variable is isolated or has coefficient 1 | Substitution |
| Coefficients of a variable match or are opposites | Elimination |
| Both equations are in standard form | Elimination |
| Fractional or decimal coefficients are present | Elimination after clearing denominators |
Either method returns the same solution set. The choice affects effort, not the answer.
5. Larger systems and Gaussian elimination
For \(n\) equations in \(n\) variables, elimination generalizes to Gaussian elimination. Write the augmented matrix, use row operations to reach row echelon form, then back-substitute. The allowed row operations are the matrix form of the three equivalence operations above [3].
See also
References
- ^OpenStax, College Algebra 2e, Rice University (Systems of linear equations: two variables).
- ^James Stewart, Lothar Redlin, and Saleem Watson, Precalculus: Mathematics for Calculus, 7th ed., Cengage Learning, 2015.
- ^Gilbert Strang, Linear Algebra and Its Applications, 5th ed., Cengage Learning, 2016.