Key takeaway: The elimination method makes one unknown disappear by combining two equations, adding or subtracting them, so only one unknown is left to solve for.
Grade 88.EE.C.8bExpressions and Equations
Definition of Elimination Method
The elimination method exploits a simple fact: adding equal things to equal things keeps the balance. Stack two equations, add or subtract them, and if the coefficients of one unknown are opposites or equal, that unknown vanishes.
Look at the y-terms in \(3x + 2y = 17\) and \(5x – 2y = 7\). They are \(+2y\) and \(-2y\), so adding the equations makes them cancel. When no pair cancels, multiply one or both equations by a number first to create matching coefficients. The method shines when both equations are written in the same \(Ax + By = C\) pattern, and it avoids fractions that substitution can produce.
Formula and notation
Line up like terms, then combine the left sides and the right sides together:
\[(3x + 2y) + (5x – 2y) = 17 + 7 \ \Rightarrow\ 8x = 24\]
Opposite coefficients: add. Equal coefficients: subtract.
Key facts
- The elimination method adds the equations when coefficients of one unknown are opposites.
- Subtract the equations when the coefficients are exactly equal.
- If nothing matches, multiply a whole equation to create a match.
- After finding one variable, substitute to get the other and check.
Elimination Method example
Use the elimination method on \(3x + 2y = 17\) and \(5x – 2y = 7\).
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The y-terms are \(+2y\) and \(-2y\), opposites, so add: \(8x = 24\) and \(x = 3\).
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Substitute into the first equation: \(3(3) + 2y = 17\), so \(2y = 8\) and \(y = 4\).
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Check in the second: \(5(3) – 2(4) = 15 – 8 = 7\). It works.
The solution is \((3, 4)\).
Common mistake
Watch out
For \(4x + 3y = 18\) and \(2x + 3y = 12\), Beck adds the equations and gets \(6x + 6y = 30\), which still has two unknowns.
Why it is wrong
The y-coefficients are equal, not opposite, so adding keeps them. Subtract instead: \((4x – 2x) + (3y – 3y) = 18 – 12\), giving \(2x = 6\) and \(x = 3\). Then \(12 + 3y = 18\) gives \(y = 2\). Remember to subtract every term, including the right-hand sides.
Related terms and lessons
- Grade 8 · Systems of Linear EquationsLessonWorksheetQuiz
- Algebra 1 · Systems of Linear EquationsLessonWorksheetQuiz
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