Algebra IV (AQA GCSE Further Maths): Revision Notes
Simultaneous equations in three unknowns
What are simultaneous equations in three unknowns?
When you have three different variables (usually called , , and ) that appear in multiple equations, you're dealing with simultaneous equations in three unknowns. To find unique values for all three variables, you need exactly three separate equations that contain these variables.
Before tackling these problems, you should already feel confident solving simultaneous equations with just two unknowns, as this skill forms the foundation for the more complex three-variable problems.
The four-step elimination method
There's a systematic approach to solving these equations that makes the process much more manageable. Here's the method broken down into clear steps:
Step 1: Choose one variable to eliminate first, then combine two of your equations to remove that variable completely.
Step 2: Take the same variable you eliminated in Step 1, and eliminate it again by combining a different pair of equations.
Step 3: You now have two equations with only two unknowns remaining. Solve this pair using the standard methods for two-variable simultaneous equations.
Step 4: Once you've found the values for two variables, substitute these back into any of your original three equations to find the value of the third variable.
The key to success is being systematic and eliminating the same variable in both Steps 1 and 2.
Worked example: using multiplication and addition
Let's work through a complete example to see how this method works in practice.
Worked Example: Multiplication and Addition Method
Problem: Solve the system:
- (equation ①)
- (equation ②)
- (equation ③)
Step 1: Let's eliminate by combining equations ① and ②.
- Multiply equation ① by 4:
- Multiply equation ② by 3:
- Add these together: (equation ④)
Step 2: Now eliminate again using a different pair.
- Multiply equation ③ by 2:
- Keep equation ② as is:
- Subtract the second from the first: (equation ⑤)
Step 3: Solve the two-variable system of equations ④ and ⑤.
- From equations ④ and ⑤: and
- Multiply equation ④ by 3:
- Subtract equation ④ from this:
- Therefore:
Substitute into equation ④: , so
Step 4: Find by substituting and into equation ①.
Solution: x = 1, y = -2, z = 4
Worked example: rearranging equations first
Sometimes the equations aren't in standard form, so you'll need to rearrange them first before applying the elimination method.
Worked Example: Rearranging Equations First
Problem: Solve the system:
Step 1: First, rearrange all equations into standard form:
- (equation ①)
- becomes (equation ④)
- becomes (equation ⑤)
Step 2: Now we can eliminate variables systematically. From the rearranged equations, we get equation ④: and equation ⑤: .
Step 3: Solve this two-variable system:
- From equation ⑤:
- Substitute into equation ④:
Find :
Step 4: Find using equation ①:
Solution: x = 3, y = -4, z = 2
Common exam tips and problem-solving strategies
When approaching these problems in exams, remember these key strategies:
Stay organised: Label your equations clearly (①, ②, ③, etc.) and number your steps to keep track of your progress.
Choose wisely: Pick the variable that's easiest to eliminate first - look for coefficients that are already the same or simple multiples of each other.
Check your work: Always substitute your final answers back into all three original equations to verify they're correct. This catches calculation errors and confirms your solution.
Show working clearly: Examiners want to see your method even if you make a calculation error. Clear working can earn you partial marks.
The elimination method is systematic and reliable. Even if the algebra gets messy, following these four steps will always lead you to the solution.
Key Points to Remember:
- You need exactly three equations to solve for three unknowns uniquely
- Always eliminate the same variable in Steps 1 and 2 of the method
- The four-step process is: eliminate, eliminate, solve two-variable system, substitute back
- Keep your working organised and clearly labelled throughout
- Check your final answer by substituting back into all original equations