How to Solve Systems of Equations: 3 Methods with Examples

A system of equations is a set of two or more equations with the same variables. Solving the system means finding values that satisfy all equations simultaneously. There are three main methods: substitution, elimination, and matrices (Cramer's rule). Each has its strengths, and the right choice depends on the structure of the problem.

Method 1: Substitution

The substitution method works by solving one equation for one variable, then substituting that expression into the other equation. It is particularly useful when one equation is already solved for a variable, or when a coefficient is 1 or -1.

Substitution Example 1

Problem: Solve the system:

Since the first equation already gives in terms of , substitute for in the second equation:

Substitute back: . The solution is .

Substitution Example 2

Problem: Solve the system:

Solve the first equation for : we get . Substitute into the second equation:

Then . The solution is .

Method 2: Elimination

The elimination method (also called the addition method) works by adding or subtracting equations to eliminate one variable. It is ideal when the coefficients of one variable are already equal or can easily be made equal by multiplication.

Elimination Example 1

Problem: Solve the system:

The coefficients are and . Adding the equations eliminates :

Substitute into the first equation: . The solution is .

Elimination Example 2

Problem: Solve the system:

Multiply the first equation by to get , then add to the second equation:

Substitute into the first equation: . The solution is .

Method 3: Matrices and Cramer's Rule

Cramer's rule uses determinants to find each variable. For a system:

The determinant of the coefficient matrix is:

And the solutions are:

Cramer's rule only works when (i.e. the system has a unique solution).

Cramer's Rule Example 1

Problem: Solve using Cramer's rule:

Calculate the determinants and solve:

The solution is .

Cramer's Rule Example 2

Problem: Solve using Cramer's rule:

The solution is .

When to Use Each Method

• Substitution: Best when one equation is already solved for a variable, or a coefficient is 1 or -1.
• Elimination: Best when coefficients of one variable are equal or opposite across equations. Very efficient for and systems.
• Cramer's rule: Best for a systematic, formulaic approach. Scales to larger systems but becomes expensive beyond .

Special Cases

Not every system has a unique solution. If the equations represent parallel lines, there is no solution (the system is inconsistent). This occurs when and elimination produces a contradiction like . If the equations represent the same line, there are infinitely many solutions (the system is dependent). Here as well, but elimination produces .

Try It Yourself

Ready to solve your own systems? Use our free System of Equations Calculator to solve systems of two or three equations instantly using any method. Enter your coefficients and see the step-by-step solution. For the matrix approach, our Matrix Calculator can handle row reduction and inverse calculations.