The Essential Guide to Non-Linear Simultaneous Equations
If you’re new to Additional Math (A Math), non-linear simultaneous equations can seem tricky at first. Most of the time, these types of questions form the base of many foundational Secondary 3 & 4 Additional Math questions, especially in the O-Levels as well – which makes it essential for you to know how to master them!
At the end of this article, you’ll learn step-by-step methods to solve different types of non-linear simultaneous equations, along with practice questions and detailed explanations! By following the examples and techniques here, you’ll be able to approach exam questions systematically and understand the logic behind each solution.
What Are Non-Linear Simultaneous Equations?
Typically, there are 2 types of simultaneous equations – Linear and Non-linear. Though you might be quite familiar with the former, here’s where it gets confusing when the equations given are no longer as simple as before with only 1 solution/intersection.
Let’s start off with a quick recap: When solving simultaneous equations, the solutions of the pair of linear and/or non-linear equations are the coordinates of the intersection point(s) of the two graphs.
Does this sound familiar to you?
Step-by-Step Method to Solve Non-Linear Simultaneous Equations
Now that we’ve had an introduction to the types of simultaneous equations and what they entail, here’s a step-by-step guide to how you can solve non-linear equations easily.
Step 1 | Make a variable the subject using the linear equation |
Step 2 | Substitute the variable into the quadratic equation such that the equation ends up with only 1 variable |
Step 3 | Solve the quadratic equation |
Step 4 | Using the value found, substitute to find the other unknown value |
With these four simple steps in mind, you can now approach any non-linear simultaneous equation methodically. By breaking the problem down, substituting carefully, and solving step by step, you’ll be able to find the correct solutions without feeling overwhelmed.
See the Steps in Action: Here’s A Worked Example
Next, let’s work our way through a sample question and solution together! This will help to put things into perspective and allow you to visualise these steps better.
Example #1
When we read this question, we can then infer that there can be a maximum of 2 solutions resulting from the intersecting points between the straight line and the curve.
By following our 4-step method as described previously, we make y the subject in this case (note that it’s easier to make y the subject as its coefficient is 1) and labelled our new equations as 1 and 2. This then makes it simpler for us to substitute accordingly. As we end up with 2 values for variable x, we just have to substitute each value back into equation 1 to obtain our final coordinates as our answers.
Example #2
In our previous example, the equations to be solved were clearly stated in the question. However, there may be cases where we have to apply this concept to more situational questions instead – here’s another example!
Question: Ann is older than her sister Betty. Their ages in years are such that twice the square of Betty’s age subtracted from the square of Ann’s age gives a number equal to 6 times the difference of their ages. Given also that the sum of their ages is equal to 5 times the difference of their ages, find the age in years of each of the sister.
Test Yourself: Try Solving These Equations
We hope that our worked example gave you a better understanding of a stepwise approach to solving these equations. Next, let’s put those skills into practice!
Practice Question #1
Check your answers here: x = 3, y = 1 and x = 1, y = 5
Practice Question #2
Check your answers here: Area of Triangle is 630cm2
Turn Knowledge into Results — Start Practising Today!
Tackling topics like non-linear simultaneous equations doesn’t have to feel overwhelming. With the strategies and step-by-step approach you’ve just learned, you can approach these problems methodically, understand the underlying patterns, and solve them with confidence.
If you want to take your skills further, our Secondary Math Trial Class gives you guided practice with exam-style questions, personalised tips, and proven strategies to handle even trickier questions. Experience how structured support can make challenging Secondary 3 & 4 A Math topics more approachable — and start turning practice into results today!
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