Elimination Method
In short: Elimination method solves a system by adding or subtracting the equations so that one variable cancels, leaving one equation in a single unknown.
8th GradeAlgebra8.EE.C.8b
What is Elimination Method?
The elimination method exploits a simple fact: adding equal amounts to both sides of an equation keeps it true. Add two equations of a system together and, if one variable has opposite coefficients, that variable vanishes.
In \(2x + 3y = 23\) and \(4x – 3y = 1\), the terms \(+3y\) and \(-3y\) cancel when you add, giving \(6x = 24\). When no variable cancels at first, multiply one or both equations by a number to create matching or opposite coefficients.
How to spot it
- The elimination method fits when both equations have x and y terms on the left and a number on the right.
- One variable has equal or opposite coefficients in the two equations.
- No variable is isolated, so substitution would involve awkward fractions.
- Multiplying one equation by a small whole number would line up a pair of coefficients.
The method
- To run the elimination method, line up the equations so like terms sit in the same columns.
- Multiply one or both equations so a variable has opposite coefficients.
- Add the equations to eliminate that variable, and solve for the other one.
- Substitute the value into either original equation to find the remaining variable, then check both.
Elimination Method: solved example
Problem
Solve the system \(2x + 3y = 23\) and \(4x – 3y = 1\) by elimination.
Solution
The y-terms are \(3y\) and \(-3y\), so add the equations: \(6x = 24\) and \(x = 4\).
Put \(x = 4\) into the first: \(8 + 3y = 23\), so \(y = 5\). Check the second: \(16 – 15 = 1\). The solution is \((4, 5)\).
Try it
Question 1
Solve \(x + 2y = 11\) and \(3x – y = 12\). Hint: multiply the second equation by 2 first.
Show the answer
The second becomes \(6x – 2y = 24\). Add to the first: \(7x = 35\), so \(x = 5\); then \(5 + 2y = 11\) gives \(y = 3\).
Question 2
In the example, why was addition used instead of subtraction?
Show the answer
The y coefficients, 3 and -3, were opposites, so adding made them cancel; subtracting would have doubled the y terms.
Elimination Method vs. Substitution Method
Elimination combines whole equations to cancel a variable, and works best when both equations are lined up in the form Ax + By = C. Substitution plugs an expression into the other equation, and works best when one variable is already alone. Elimination also makes it easy to spot systems with no solution: both variables vanish and a false statement remains.
Keep going
Lessons
Worksheets and quizzes
- Systems of Linear Equations – Worksheet
- Systems of Linear Equations – Quiz
- Systems of Linear Equations – Worksheet
- Systems of Linear Equations – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Algebra