Ch 5 · 5 Simultaneous Equations · O Level Additional Mathematics 4037
Topic 5 of 6 in Algebra and Functions — O Level Additional Mathematics 4037 — 44 lessons.
Introduction
Where does a straight path cross a curved fence? When do two growing quantities become equal? Both questions mean solving two equations at once. In Additional Mathematics one equation is often a quadratic or a product such as xy^2 = 4, so substitution is the main tool. (Syllabus topic 5.)
Key ideas — Elimination
When both equations are linear (or can be made linear, e.g. by taking logs or using index laws), multiply them so that one unknown has the same coefficient, then add or subtract.
🧑🏫 Teacher's explanation
Imagine you are trying to balance two different scales. Each scale represents an equation. If you have two scales that are perfectly balanced, and you double the weights on both, they stay balanced. If you subtract the weights of one scale from the other, the remaining difference must also be balanced. In algebra, 'elimination' is the process of making one variable 'disappear' by adding or subtracting the equations, allowing you to focus on the one that remains.
📖 Definitions
🔍 Worked example — Easy
Solve the simultaneous equations: 3x + 2y = 16 x - y = 2
🔍 Worked example — Standard
Solve the simultaneous equations: 5x - 3y = 1 2x + 4y = 22
🔍 Worked example — Exam-style
Solve the simultaneous equations (one non-linear): x + y = 7 x^2 + y^2 = 25
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
2x + 3y = 13 $x^2 - y^2 = 5 [4 marks]
Key ideas — Substitution
When one equation is linear and the other is not:
🧑🏫 Teacher's explanation
Imagine you are trying to find the exact point where a straight road crosses a curved river. The straight road is your linear equation, and the curved river is your non-linear equation. To find the intersection, you need to find the values of x and y that satisfy both at the same time.
📖 Definitions
🔍 Worked example — Standard
Solve the simultaneous equations: x - y = 3 x^2 + 2y^2 = 22
🔍 Worked example — Exam-style
Solve the simultaneous equations: y + x = 4 xy = 3
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
2x - y = 1 x^2 + y^2 = 10 [4 marks]
Key ideas — Other non-linear pairs
🧑🏫 Teacher's explanation
When you face simultaneous equations where variables are multiplied together (like xy) or raised to powers, the standard 'subtraction' method used in linear equations often fails. Instead, think of it like a puzzle where you want to isolate one variable by 'cancelling' the other. If you have xy^2 = 4 and xy = 3, you can see that the first equation is just the second equation multiplied by another y. By dividing the two equations, that y term disappears, leaving you with a simple linear equation for y.
📖 Definitions
🔍 Worked example — Easy
Solve the simultaneous equations: xy^2 = 4 and xy = 3
🔍 Worked example — Standard
Solve the simultaneous equations: 2^x × 4^y = 32 and x - y = 1
🔍 Worked example — Exam-style
Solve the simultaneous equations: x^2y = 12 and xy = 4
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
3^x+y = 81 $x - y = 2 [4 marks]
Key ideas — Geometric meaning
The solutions are the intersection points of the two graphs. A quadratic with two roots means two points; a repeated root means the line is a tangent; no real roots means the graphs do not meet (topic 2).
🧑🏫 Teacher's explanation
When you solve simultaneous equations, you aren't just finding numbers that satisfy two rules; you are finding the exact locations where two paths cross. Imagine you are walking on a straight path (a linear equation) and a friend is walking on a curved path (a non-linear equation). The solutions to the equations are the specific coordinates (x, y) where you and your friend bump into each other.
📖 Definitions
| Term | Cambridge wording | | --- | --- | | Intersection | A point where two or more curves or lines meet. | | Tangent | A straight line that touches a curve at a single point without crossing it at that point. | | No real solutions | A situation where the graphs of the equations do not intersect in the real Cartesian plane. |
🔍 Worked example — Standard
Determine the number of intersection points between the line y = 2x + k and the curve y = x^2.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
Solve y - x + 3 = 0 and x^2 - 3xy + y^2 + 19 = 0.
🔍 Worked example — Easy
Solve y - x + 3 = 0 and x^2 - 3xy + y^2 + 19 = 0.
🔍 Worked example — Standard
Solve xy^2 = 4 and xy = 3.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) x = 7 - 2y: 5y^2 - 28y + 39 = 0, y = 3 or 13/5: (1, 3) and (9/5, 13/5 ). (2) x(2x - 1) = 3 ⇒ 2x^2 - x - 3 = 0: x = tfrac32,\ y = 2 or x = -1,\ y = -3. (3) x^2 + x(x + 1) = 6 ⇒ 2x^2 + x - 6 = 0: (tfrac32, tfrac52 ) and (-2, -1). (4) x + y = 3, x - y = -1: x = 1, y = 2.