Substitution, Elimination, Addition and Equalization Methods
The algebraic methods for solving a system of linear equations are substitution, elimination (addition) and equalization, and each reduces the system to one equation with one unknown. In brief: to solve x + y = 7 and 2x − y = 2, add the equations to eliminate y: 3x = 9, so x = 3 and y = 4. Substitution and equalization reach the same answer by a different route.
All three methods are exact and work well when the graphical method would only give an estimate. Which one is best depends on the form of the equations. This guide explains each method with a worked example, shows how to choose and gives a comparison table.
How does the substitution method work?
- Solve one equation for one unknown, preferably one with coefficient 1.
- Substitute the expression into the other equation.
- Solve the resulting equation with one unknown.
- Substitute back to find the second unknown.
- Check in both equations.
Example: y = 2x − 1 and 3x + 2y = 12
The first equation already gives y. Substitute into the second: 3x + 2(2x − 1) = 12. Expand: 3x + 4x − 2 = 12, so 7x = 14 and x = 2. Then y = 2 · 2 − 1 = 3. Check: 3 · 2 + 2 · 3 = 12 ✓. The solution is (2, 3).
How does the equalization method work?
Solve both equations for the same unknown and set the two expressions equal. This is a special case of substitution and is natural when both equations are in the form y = ….
Example: y = 3x − 4 and y = −x + 8
Set 3x − 4 = −x + 8, so 4x = 12 and x = 3. Then y = 3 · 3 − 4 = 5. Check with the second: −3 + 8 = 5 ✓. The solution is (3, 5).
How does the elimination (addition) method work?
- Write both equations in the form ax + by = c, with the unknowns aligned.
- Multiply one or both equations so that one unknown has opposite coefficients.
- Add the equations. That unknown cancels.
- Solve for the remaining unknown and substitute back.
Example 1: coefficients already opposite
x + y = 7 and 2x − y = 2. Adding gives 3x = 9, so x = 3, and y = 7 − 3 = 4. Check: 2 · 3 − 4 = 2 ✓.
Example 2: multiplying one equation
3x + 2y = 16 and x − y = 2. Multiply the second by 2: 2x − 2y = 4. Add to the first: 5x = 20, so x = 4. Then y = x − 2 = 2. Check: 12 + 4 = 16 ✓.
Example 3: multiplying both equations
2x + 3y = 13 and 5x − 2y = 4. Multiply the first by 2 and the second by 3: 4x + 6y = 26 and 15x − 6y = 12. Add: 19x = 38, so x = 2. Substituting: 4 + 3y = 13, so y = 3. Check: 10 − 6 = 4 ✓.
Which method should you choose?
- If one equation already has a variable isolated, such as y = 2x − 1, use substitution.
- If both equations are in the form y = …, use equalization.
- If both are in standard form ax + by = c, use elimination.
Elimination is the method that extends to larger systems. If during the process a variable disappears and a false statement remains, such as 0 = 5, the system has no solution. A true statement 0 = 0 means infinitely many solutions.
Comparison of the three methods
| Method | Best for | Key step |
|---|---|---|
| Substitution | One variable already isolated | Replace the variable by its expression |
| Equalization | Both equations solved for the same variable | Set the expressions equal |
| Elimination | Standard form ax + by = c | Add multiples to cancel a variable |
Where do the methods come from?
Elimination is the oldest: the Chinese text The Nine Chapters on the Mathematical Art, compiled around the first century, describes a rule that is essentially elimination on an array of numbers. The method was later named after Carl Friedrich Gauss, although he did not invent it, and the substitution approach was a standard part of algebra from the introduction of symbolic notation.
Where are they used today?
Computers solve systems with thousands of equations using elimination-style algorithms, for example in structural engineering and weather models. For small problems by hand, such as mixing two solutions, finding two unknown prices or balancing a budget, substitution or elimination is usually enough.
Which mistakes are common?
Multiplying only one side
When you multiply an equation by a number, multiply every term, including the right side.
Adding when you should subtract
If the coefficients of the unknown have the same sign, subtract the equations or multiply one by −1.
Stopping after the first variable
Substitute back to find the second unknown, then check in both original equations.
Sign mistakes in substitution
When substituting an expression with a minus sign, use brackets: 3x + 2(2x − 1), not 3x + 2 · 2x − 1.