Solve systems of two linear equations with two variables using the elimination method.
Last updated: March 2026 | By ForgeCalc Engineering
The elimination method (also known as the addition method) is a technique for solving systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, leaving an equation with only one variable to solve.
This method is often more efficient than substitution, especially when the coefficients are integers that can be easily scaled to match.
Solve: (1) 2x + y = 5, (2) x - y = 1
1. Add Eq 1 and Eq 2: (2x + x) + (y - y) = 5 + 1
2. Result: 3x = 6
3. Solve for x: x = 2
4. Substitute x=2 into Eq 2: 2 - y = 1 ⇒ y = 1
Final Answer: (2, 1)
Elimination is usually better when neither equation has a variable with a coefficient of 1 or -1, as substitution would involve working with fractions early on.
Yes. You can choose whichever variable is easier to eliminate based on the coefficients.
If they are the same (e.g., 2x and 2x), subtract the equations. If they are opposites (e.g., 2x and -2x), add them.
Yes, but you have to apply elimination twice to reduce the system to 2 variables, then solve that system.
Find the least common multiple (LCM) of the coefficients you want to eliminate. Multiply the first equation by (LCM / coefficient₁) and the second by (LCM / coefficient₂).
If you eliminate one variable and get an impossible statement (like 0 = 5), the lines are parallel and never intersect. The system is inconsistent.
If you eliminate one variable and get a true statement (like 0 = 0), the two equations describe the same line. There are infinitely many solutions.
Elimination works because adding or subtracting equivalent equations preserves the solution set. If (a, b) satisfies both original equations, it also satisfies the sum or difference of those equations.
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