Elimination method
Solve simultaneous equations by adding or subtracting the equations to eliminate one variable. When coefficients don't match, multiply one or both equations first.
Worked examples
Direct subtraction eliminates a variable
Straightforward
Problem
Solve the simultaneous equations:
1
The -coefficients are both 2. Subtract the second equation from the first to eliminate .
2
Solve for .
3
Substitute into the simpler equation to find .
4
Check in both original equations.
Equation 1: ✓
Equation 2: ✓
Equation 2: ✓
Answer
,
Multiply one equation to match coefficients
Moderate
Problem
Solve the simultaneous equations:
1
Multiply the second equation by so the -coefficients match in size.
2
Add the modified second equation to the first equation to eliminate .
3
Solve for .
4
Substitute into the second original equation to find .
5
Check in both original equations.
Equation 1: ✓
Equation 2: ✓
Equation 2: ✓
Answer
,
Multiply both equations
Challenging
Problem
Solve the simultaneous equations:
1
No coefficients match. Multiply the first equation by and the second by to make the -coefficients equal in size.
2
Add the two new equations to eliminate .
3
Solve for .
4
Substitute into the first original equation to find .
5
Check in both original equations.
Equation 1: ✓
Equation 2: ✓
Equation 2: ✓
Answer
,
Practise
Q1·Straightforward
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Add the equations to eliminate :
Substitute into the first equation:
Check: ✓
Solution: ,
Substitute into the first equation:
Check: ✓
Solution: ,
Q2·Straightforward
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Subtract the second equation from the first:
Substitute into the second equation:
Check: ✓
Solution: ,
Substitute into the second equation:
Check: ✓
Solution: ,
Q3·Straightforward
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Subtract the second from the first:
Substitute into the second equation:
Check: ✓
Solution: ,
Substitute into the second equation:
Check: ✓
Solution: ,
Q4·Straightforward
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Subtract the second from the first:
Substitute into the second equation:
Check: ✓
Solution: ,
Substitute into the second equation:
Check: ✓
Solution: ,
Q5·Moderate
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Multiply the second equation by :
Subtract from the first equation:
Substitute into :
Check: ✓
Solution: ,
Subtract from the first equation:
Substitute into :
Check: ✓
Solution: ,
Q6·Moderate
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Multiply the second equation by :
Add to the first equation:
Substitute into :
Check: ✓
Solution: ,
Add to the first equation:
Substitute into :
Check: ✓
Solution: ,
Q7·Moderate
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Add the equations (the -terms cancel):
Substitute into :
Check: ✓
Solution: ,
Substitute into :
Check: ✓
Solution: ,
Q8·Moderate
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Add the equations:
Substitute into :
Check: ✓
Solution: ,
Substitute into :
Check: ✓
Solution: ,
Q9·Challenging
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Multiply the first equation by :
Multiply the second equation by :
Subtract:
Substitute into :
Check: ✓
Solution: ,
Multiply the second equation by :
Subtract:
Substitute into :
Check: ✓
Solution: ,
Q10·Challenging
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Multiply the first equation by :
Multiply the second equation by :
Add ( is eliminated):
Substitute into :
Check: ✓
Solution: ,
Multiply the second equation by :
Add ( is eliminated):
Substitute into :
Check: ✓
Solution: ,
Q11·Challenging
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Multiply the first equation by :
Multiply the second equation by :
Subtract:
Substitute into :
Check: ✓
Solution: ,
Multiply the second equation by :
Subtract:
Substitute into :
Check: ✓
Solution: ,
Q12·Challenging
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Explanation
Multiply the first equation by :
Add to the second equation:
Substitute into :
Check: ✓
Solution: ,
Add to the second equation:
Substitute into :
Check: ✓
Solution: ,
Open Math
Elimination method
Algebra · MA5-EQU-P-01
Name:
Date:
Q1Straightforward
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Q2Straightforward
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Q3Straightforward
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Q4Straightforward
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Q5Moderate
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Q6Moderate
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Q7Moderate
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Q8Moderate
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Q9Challenging
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Q10Challenging
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Q11Challenging
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Q12Challenging
Solve the simultaneous equations using elimination.
Enter your answer as (x, y).
Enter your answer as (x, y).
Worked solutions and answers at openmath.au/year-9/simultaneous-equations/elimination-method