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: ,