Elimination Method for Solving Simultaneous Equations
The elimination method removes one variable from simultaneous equations by adding or subtracting the two equations. The goal is to cancel one variable, leaving a single equation like 2x = 12 to solve.
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Elimination Method
- The elimination method removes one variable by adding or subtracting two equations.
- The aim is to get an equation like that is easy to solve.
Elimination by Addition
- Add the equations when one variable has opposite signs, like +y and −y.
- That variable cancels out, leaving an equation with one variable.
Preparing the Equations
- Transform the equations so one variable can be cancelled, like +3y and −3y.
- Exam tip: when you multiply, multiply every term on both sides of the equation.
Elimination by Subtraction
- Subtract the equations when one variable has the same sign, like +2x and +2x.
- This removes that variable so you can solve.
Finishing and Checking
- Substitute the value back to find the other variable.
- Exam tip: always check by substituting both x and y into the original equations.
Practice Questions
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Solve the simultaneous equations:
Correct! 🎉 +10 pointsNot quite right
Add the two equations together: . This eliminates and simplifies to , so . Substituting into either equation gives , which simplifies to . Therefore, the solution is and .
Solve the simultaneous equations:
Correct! 🎉 +10 pointsNot quite right
Add the two equations together: . This simplifies to , so . Substituting into gives , so . Therefore, the solution is and .
Solve the simultaneous equations:
Correct! 🎉 +20 pointsNot quite right
Add the two equations together: . This eliminates and simplifies to , so . Substituting into gives , which simplifies to and . Therefore, the solution is and .
Solve the simultaneous equations:
Correct! 🎉 +20 pointsNot quite right
Subtract the second equation from the first: . This simplifies to , so . Substituting into gives , which simplifies to . Therefore, the solution is and .
Solve the simultaneous equations:
Correct! 🎉 +20 pointsNot quite right
Subtract the first equation from the second: . This eliminates and simplifies to , so . Substituting into gives , which simplifies to and . Therefore, the solution is and .
Solve the simultaneous equations:
Correct! 🎉 +30 pointsNot quite right
First multiply the second equation by 2 to align the coefficients of : . Now add this to the first equation: . This simplifies to , so . Substituting into gives , which simplifies to and . Therefore, the solution is and .
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Solve simultaneous equations step-by-step using the elimination method
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Students Also Ask
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The elimination method solves simultaneous linear equations: two equations that share the same two unknowns, such as x and y. You combine the equations to eliminate one unknown, solve the equation that remains, then substitute that value back to find the second unknown.
Either works, and both give the same correct solution. Add the equations when the matching terms have opposite signs, because they then cancel to zero. Subtract them when the matching terms have the same sign. Pick whichever route removes a variable most simply for your equations.
Transform the equations first. Multiply one of the equations by a suitable number so that a pair of coefficients match in size. Once those terms match, you can add or subtract the equations to eliminate that variable, then carry on solving as normal.
Adding works when one variable has equal and opposite terms in the two equations, such as +y in one and -y in the other. When you add the equations, that pair sums to zero and disappears, leaving a single equation with just one unknown to solve.
Substitute both values back into the original equations, not the transformed ones. Work out each side and confirm both balance. For example, putting x = 4 and y = 2 into 3x + 2y gives 12 + 4 = 16, which is correct, so the solution checks out.
Two slips are common. The first is trying to add or subtract before a pair of coefficients match, so nothing cancels; multiply an equation first so the terms line up. The second is forgetting the final check, so always substitute both values back into the original equations.