LEARNING OBJECTIVES
What you will be able to do
- choose between elimination and substitution
- solve linear–nonlinear systems
- handle product and reciprocal relationships
- check solutions and interpret intersections
AT A GLANCE
INTRODUCTION · THE BIG IDEA
Reduce two linked conditions to one equation, then recover and verify every ordered pair.
Simultaneous equations ask for values that satisfy both conditions at once. In Additional Mathematics, one or both equations may be nonlinear, so substitution often creates a quadratic or higher equation.
Each real solution for one variable may generate a matching value for the other. The answer is a set of ordered pairs, not two unrelated lists.
SECTION 01
Choosing an elimination route
Use elimination when matching terms can be added or subtracted. Use substitution when one variable, product or reciprocal expression can be isolated cleanly.
| Structure | Efficient start |
|---|---|
| two linear equations | eliminate a variable |
| one variable already isolated | substitute directly |
| xy is known | replace the repeated product |
| reciprocal expression | clear denominators after restrictions |
| line with a curve | substitute the line into the curve |
SECTION 02
Linear and quadratic systems
Substituting a line into a quadratic relation gives one quadratic equation. Its discriminant also predicts whether the graphs meet twice, touch once or do not meet in real coordinates.
Intersect a line and a parabola
- Solve y = x + 1 with x² + y² = 13.
- Substitute: x² + (x + 1)² = 13.
- Simplify: x² + x − 6 = 0, so x = 2 or x = −3.
- Use y = x + 1 to obtain y = 3 or y = −2.
Answer: (2, 3) and (−3, −2).
SECTION 03
Products, powers and reciprocals
If a relation such as xy = k is available, use y = k/x with x ≠ 0 or replace xy wherever it appears. Equations containing y/x and x/y should be multiplied by a valid common denominator.
DETAILED EXPLANATION
- If xy² and xy are known, division may reveal y.
- A substitution can produce powers such as y²; recover both signs where valid.
- Check that no multiplication by zero introduced a candidate.
SECTION 04
Checking and presenting solutions
Substitution, squaring and clearing denominators can introduce extraneous results. Test each pair in the original equations, then present paired values together.
DETAILED EXPLANATION
- Complex solutions are not required unless a question explicitly changes the number system.
- Graphical solutions should be read to the requested accuracy.
- Reject a pair if either original equation fails.
QUICK CHAPTER SUMMARY
The ideas to carry forward
- A simultaneous solution satisfies all equations.
- Elimination suits matching terms; substitution suits an isolated expression.
- Nonlinear systems can have several, one or no real pairs.
- Restrictions must be stated before clearing denominators.
- Every candidate pair should be checked in the original system.
QUICK REVISION CHECKLIST
Can you do each of these without your notes?
- choose between elimination and substitution
- solve linear–nonlinear systems
- handle product and reciprocal relationships
- check solutions and interpret intersections