Solving simultaneous equations is one of those topics that keeps coming back in Secondary school maths – from Sec 2 all the way to O-Level E-Maths and even in A-Maths word problems.
If you’re in Singapore, you already know: MOE loves testing this in many forms.
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The good news? Once you master a few clear methods, most questions follow the same patterns. In this tutorial, I’ll walk you through:
- Step-by-step methods (elimination & substitution)
- How to choose the fastest method in exams
- Typical O-Level style twists (fractions, parameters, word problems)
- Practice questions with harder variants
- Common mistakes that cause you to lose marks
And if you want instant, 24/7 practice that’s aligned to the Singapore syllabus, I’ll also show you how to use Tutorly.sg effectively. It’s a website (not an app) that thousands of students in Singapore already use, and it’s even been mentioned on Channel NewsAsia (CNA).
Step-by-step tutorial
Let’s focus on what you actually see in your Sec 2 / O-Level E-Maths paper: two equations with two unknowns.
Typical forms:
- Linear–linear (both equations are straight lines)
- Linear–nonlinear (e.g. one is linear, the other is quadratic)
We’ll start with the basics, then move to the trickier types.
1. Method 1: Elimination (your main workhorse)
You use elimination when it’s easy to make one variable cancel out.
Example 1 (basic O-Level style)
Solve:
2 x + 3 y = 12 \\ 5 x - 3 y = 3 \end{cases}$$ **Step 1: Line up the equations** Write them one under the other: 1. $2 x + 3 y = 12$ 2. $5 x - 3 y = 3$ Notice the $+3 y$ and $-3 y$. Perfect for elimination. **Step 2: Add or subtract to eliminate one variable** Add equation (1) and (2): $$(2 x + 3 y) + (5 x - 3 y) = 12 + 3$$ $3 y$ and $-3 y$ cancel: $$7 x = 15$$ So: $$x = \frac{15}{7}$$ **Step 3: Substitute back to find the other variable** Use equation (1): $2 x + 3 y = 12$ Substitute $x = \frac{15}{7}$: $$2\left(\frac{15}{7}\right) + 3 y = 12$$ $$\frac{30}{7} + 3 y = 12$$ Move $\frac{30}{7}$ to the right: $$3 y = 12 - \frac{30}{7} = \frac{84}{7} - \frac{30}{7} = \frac{54}{7}$$ So: $$y = \frac{54}{7} \div 3 = \frac{54}{21} = \frac{18}{7}$$ **Final answer**: $x = \frac{15}{7}$, $y = \frac{18}{7}$ In exams, don’t panic if you see fractions. MOE doesn’t always give nice integers. --- ### 2. Method 2: Substitution (great when a variable is already “alone”) Use substitution when one equation already has $x$ or $y$ alone, or can be made alone easily. **Example 2 (Sec 2 / O-Level)** Solve: $$\begin{cases} y = 2 x + 1 \\ 3 x + 2 y = 17 \end{cases}$$ **Step 1: Identify the easy equation** You already have $y = 2 x + 1$. That’s very convenient. **Step 2: Substitute into the other equation** In $3 x + 2 y = 17$, replace $y$ with $(2 x + 1)$: $$3 x + 2(2 x + 1) = 17$$ Expand: $$3 x + 4 x + 2 = 17$$ Combine like terms: $$7 x + 2 = 17$$ So: $$7 x = 15 \Rightarrow x = \frac{15}{7}$$ **Step 3: Find the other variable** Use $y = 2 x + 1$: $$y = 2\left(\frac{15}{7}\right) + 1 = \frac{30}{7} + \frac{7}{7} = \frac{37}{7}$$ **Final answer**: $x = \frac{15}{7}$, $y = \frac{37}{7}$ --- ### 3. How to choose: Elimination vs Substitution When you see a question in your Sec 3/4 test or O-Level paper, ask yourself: - Is any variable already alone (e.g. $y = 3 x - 2$)? - **Yes** → Use substitution, usually faster. - Can I easily make coefficients match (e.g. $2 x + 3 y$ and $4 x - 3 y$)? - **Yes** → Use elimination. - Are there many fractions or decimals? - Often elimination is cleaner if you first clear denominators. You don’t get extra marks for choosing a “fancier” method. Choose the one that gives fewer chances to make careless mistakes. --- ### 4. Fraction and decimal equations (very common in O-Levels) MOE loves to throw in fractions to test your algebra discipline. **Example 3 (with fractions)** Solve: $$\begin{cases} \frac{x}{2} + \frac{y}{3} = 5 \\ \frac{x}{4} - \frac{y}{6} = 1 \end{cases}$$ **Step 1: Clear denominators** For the first equation, LCM of 2 and 3 is 6. Multiply the whole equation by 6: $$3 x + 2 y = 30 \quad (1)$$ For the second, LCM of 4 and 6 is 12. Multiply by 12: $$3 x - 2 y = 12 \quad (2)$$ **Step 2: Eliminate** Now add (1) and (2): $$(3 x + 2 y) + (3 x - 2 y) = 30 + 12$$ $$6 x = 42 \Rightarrow x = 7$$ Substitute $x = 7$ into (1): $$3(7) + 2 y = 30 \Rightarrow 21 + 2 y = 30 \Rightarrow 2 y = 9 \Rightarrow y = \frac{9}{2}$$ **Final answer**: $x = 7$, $y = \frac{9}{2}$ **Key idea**: Clear fractions first, then use normal elimination. This is exactly the kind of manipulation you need for O-Level E-Maths Paper 1. --- ### 5. Linear–quadratic simultaneous equations (harder exam type) These show up in Sec 3/4 and O-Levels, often in coordinate geometry or word problems. **Example 4 (linear + quadratic)** Solve: $$\begin{cases} y = x + 1 \\ x^2 + y^2 = 25 \end{cases}$$ **Step 1: Substitute the linear into the quadratic** From $y = x + 1$, substitute into $x^2 + y^2 = 25$: $$x^2 + (x + 1)^2 = 25$$ Expand: $$x^2 + (x^2 + 2 x + 1) = 25$$ $$2 x^2 + 2 x + 1 = 25$$ Bring everything to one side: $$2 x^2 + 2 x + 1 - 25 = 0$$ $$2 x^2 + 2 x - 24 = 0$$ Divide by 2: $$x^2 + x - 12 = 0$$ **Step 2: Solve the quadratic** Factorise: $$x^2 + x - 12 = (x + 4)(x - 3) = 0$$ So: - $x = -4$ or $x = 3$ **Step 3: Find the corresponding $y$ values** If $x = -4$: $$y = x + 1 = -4 + 1 = -3$$ If $x = 3$: $$y = x + 1 = 4$$ **Final answers**: $(x, y) = (-4, -3)$ and $(3, 4)$ For linear–quadratic questions, substitution is almost always the way to go. --- ## Exam strategy guide Knowing the methods is one thing. Scoring well under O-Level exam pressure is another. Here’s how to approach simultaneous equations strategically. > “Access more than 1000+ past year papers to practice” > [👉 Start a paper today and test yourself like it’s the real exam.](https://tutorly.sg/app)  ### 1. Read the question type carefully You’ll typically see simultaneous equations in: - **Pure algebra questions**: “Solve the simultaneous equations …” - **Word problems**: speed, number of pens & pencils, price discounts, etc. - **Coordinate geometry**: intersection of lines / line and curve Identify which type it is first. That tells you how much time to spend and what form your final answer should take (e.g. $(x, y)$ coordinates vs context-based answer like “the number of boys is 12”). ### 2. Time management (especially for O-Level E-Maths) - For a straightforward “solve these equations” question (2–3 marks), aim for **3–4 minutes**. - For word problems (4–6 marks), you may need **6–8 minutes**, including forming the equations. If you’re stuck for more than 2 minutes just forming the equations, **move on and come back later**. Don’t let one part destroy your whole Paper 2. ### 3. Decide your method quickly When you see the equations: - If one is already $y = \dots$ or $x = \dots$ → **Substitution**. - If coefficients look like they can match easily (e.g. $2 x+3 y$ and $4 x-3 y$) → **Elimination**. - If there are many fractions → Clear denominators, then **Elimination**. Train yourself to decide in under 10 seconds. This reduces hesitation and panic. ### 4. Always check your answers (fast but effective) For O-Level, you don’t have time to fully re-do the question, but you can: - Substitute your final $x$ and $y$ quickly into **both** original equations. - If one doesn’t match, you know there’s an error somewhere. This is where a tool like [Tutorly.sg](https://tutorly.sg/ai-tutor-singapore) helps. You can key in a similar practice question, type your final answer, and let the AI tutor show you the step-by-step solution. Compare your working to see where you went wrong and fix your pattern of mistakes before the exam. ### 5. For word problems: define variables clearly MOE loves to hide simultaneous equations inside stories. The biggest trap is unclear variables. Example pattern: > A shop sells pens at \$2 each and markers at \$3 each. Ali buys 10 items and spends \$26. > How many pens and markers did he buy? Let: - $x$ = number of pens - $y$ = number of markers Then: - $x + y = 10$ (total items) - $2 x + 3 y = 26$ (total cost) From here, it’s a standard elimination question. **Exam tip**: Write your variable definitions **in words**. It can earn you method marks even if your algebra later has mistakes. --- ## Worksheet practice Use this section like a mini worksheet. Try each question on your own first, then check the outline of the solution. If you want fully worked solutions (step-by-step), you can enter similar questions into [Tutorly.sg](https://tutorly.sg/ai-tutor-singapore) and see how to solve them clearly. ### A. Basic practice (Sec 2 / early Sec 3 level) **Question 1** Solve: $$\begin{cases} 3 x + y = 10 \\ 2 x - y = 1 \end{cases}$$ *Outline of solution*: 1. Add equations to eliminate $y$: $$(3 x + y) + (2 x - y) = 10 + 1 \Rightarrow 5 x = 11$$ So $x = \frac{11}{5}$. 2. Substitute $x$ back into either equation to find $y$. --- **Question 2** Solve: $$\begin{cases} y = 4 x - 3 \\ 2 x + y = 11 \end{cases}$$ *Outline*: 1. Substitute $y = 4 x - 3$ into $2 x + y = 11$. 2. Solve for $x$, then find $y$. --- ### B. Intermediate practice (fractions & decimals) **Question 3** Solve: $$\begin{cases} \frac{2 x}{3} + \frac{y}{2} = 7 \\ \frac{x}{6} - \frac{y}{4} = 1 \end{cases}$$ *Outline*: 1. Clear denominators for each equation: - First: multiply by 6 → $4 x + 3 y = 42$ - Second: multiply by 12 → $2 x - 3 y = 12$ 2. Use elimination (add equations) to find $x$. 3. Substitute back to find $y$. --- **Question 4** Solve: $$\begin{cases} 1.5 x + 2 y = 13 \\ 0.5 x - y = 1 \end{cases}$$ *Outline*: 1. Multiply both equations by 2 to remove decimals: - $3 x + 4 y = 26$ - $x - 2 y = 2$ 2. Use elimination or substitution to solve. --- ### C. Harder exam variants (O-Level style) These are closer to what you’ll see in O-Level E-Maths Paper 2, including parameters and linear–quadratic systems. #### Question 5 (with parameter $k$) Solve for $x$ and $y$ in terms of $k$: $$\begin{cases} 2 x + ky = 8 \\ x - 3 y = 1 \end{cases}$$ *Outline*: 1. From second equation: $x = 1 + 3 y$. 2. Substitute into first: $2(1 + 3 y) + ky = 8$. 3. Simplify: $2 + 6 y + ky = 8 \Rightarrow (6 + k)y = 6$. 4. So $y = \dfrac{6}{6 + k}$ (state restriction $k \neq -6$). 5. Then $x = 1 + 3 y = 1 + \dfrac{18}{6 + k}$. This kind of “in terms of $k$” question is common in Sec 4 and O-Levels to test algebra manipulation. --- #### Question 6 (linear–quadratic) Solve: $$\begin{cases} y = 2 x - 1 \\ x^2 + xy = 15 \end{cases}$$ *Outline*: 1. Substitute $y = 2 x - 1$ into $x^2 + xy = 15$: $$x^2 + x(2 x - 1) = 15$$ 2. Expand: $x^2 + 2 x^2 - x = 15 \Rightarrow 3 x^2 - x - 15 = 0$. 3. Solve the quadratic (factorisation or formula). 4. For each $x$ value, find the corresponding $y$ using $y = 2 x - 1$. 5. State both pairs $(x, y)$. --- #### Question 7 (word problem – number of items) A bookshop sells files at \$3 each and notebooks at \$2 each. A student buys 9 items and spends \$22. (a) Form two simultaneous equations. (b) Hence, find how many files and notebooks the student bought. *Outline*: Let: - $x$ = number of files - $y$ = number of notebooks (a) - Total items: $x + y = 9$ - Total cost: $3 x + 2 y = 22$ (b) Use elimination: - From $x + y = 9$, get $y = 9 - x$. - Substitute into $3 x + 2 y = 22$: $$3 x + 2(9 - x) = 22$$ - Solve for $x$ (files), then find $y$ (notebooks). --- #### Question 8 (harder word problem – speed) Two cyclists, A and B, travel between Town P and Town Q, a distance of 60 km. - Cyclist A travels from P to Q at $x$ km/h and takes 3 hours less than Cyclist B. - Cyclist B travels at $(x - 10)$ km/h. (a) Write down an expression for the time taken by each cyclist. (b) Form a pair of simultaneous equations in $x$ and the time taken by B. (c) Hence, find the value of $x$. *Outline*: (a) - Time taken by A: $\dfrac{60}{x}$ - Time taken by B: $\dfrac{60}{x - 10}$ Given A takes 3 hours less than B: $$\frac{60}{x} = \frac{60}{x - 10} - 3$$ This is already one equation in $x$. > “Doing Secondary Science? Pick a topic and practise like it’s a real exam — with clear answers right after.” > [👉 Try Tutorly now and start a Science topic in seconds.](https://tutorly.sg/app)  You can also introduce a second variable $t$ (time taken by B), then relate it: - $t = \dfrac{60}{x - 10}$ - $t - 3 = \dfrac{60}{x}$ From there, you can form two equations in $x$ and $t$. Then eliminate $t$ to get a quadratic in $x$. This is the kind of multi-step question where practice really matters. If you struggle with forming the equations, try similar word problems on [Tutorly.sg](https://tutorly.sg/ai-tutor-singapore) and let the AI tutor walk you through the reasoning. --- ### D. How to use [Tutorly.sg](https://tutorly.sg/app) for extra practice If you’re serious about improving before your Sec 3/4 exams or O-Levels, you need **consistent** practice, not just last-minute cramming. On [Tutorly.sg](https://tutorly.sg/ai-tutor-singapore): - You can ask it to generate **new simultaneous equations questions** at your level (Sec 2 / Sec 3 / O-Level). - After you try the question on your own, type in your **final answer**. - If it’s wrong, Tutorly will show you a clear, step-by-step worked solution so you can see exactly how to do it. - You can then ask for **similar questions** to drill your weak spots (e.g. “more with fractions”, “more linear-quadratic ones”, “more word problems”). Because it’s available 24/7 as a website, you can revise whenever you have a bit of free time – between CCA, tuition, or on weekends. --- ## Common mistakes Even strong students lose marks on simultaneous equations because of small, repeated errors. Here are the big ones to avoid. ### 1. Sign errors when adding/subtracting equations Example: $$\begin{cases} 2 x + 3 y = 11 \\ 4 x - 3 y = 5 \end{cases}$$ If you add them: $$(2 x + 3 y) + (4 x - 3 y) = 11 + 5 \Rightarrow 6 x = 16$$ Some students accidentally write $6 x + 6 y = 16$ because they forget that $+3 y$ and $-3 y$ cancel. **Fix**: - Write the equations neatly, line up $x$ and $y$ terms. - Use a light pencil mark to show which terms are cancelling. --- ### 2. Forgetting to substitute back to find the second variable Sometimes you solve for $x$ and then rush to the next question, forgetting to find $y$. **Fix**: Train yourself: every time you get one variable, **immediately** write “Find $y$:” (or $x$) and do it. Make it a habit. --- ### 3. Messy handling of fractions When you have: $$\frac{x}{2} + \frac{y}{3} = 4$$ Some students try to do elimination directly and get lost. **Fix**: 1. Always clear denominators first: - LCM of 2 and 3 is 6 → multiply whole equation by 6: $$3 x + 2 y = 24$$ 2. Then treat it like a normal integer-coefficient equation. --- ### 4. Wrong substitution (mixing up expressions) Example: $$\begin{cases} y = 3 x - 2 \\ 2 x + y = 10 \end{cases}$$ Some students copy wrongly as $y = 3 - 2 x$ or substitute $y = 3 x + 2$. **Fix**: - When substituting, **circle** the expression you’re replacing. - Rewrite the second equation with brackets: $2 x + (3 x - 2) = 10$. - Expand step by step. --- ### 5. Not checking for “no solution” or “infinite solutions” Sometimes, after elimination, you may get: - $0 = 5$ → **No solution** (inconsistent equations, lines are parallel) - $0 = 0$ → **Infinitely many solutions** (same line) MOE occasionally tests this concept. **Example**: $$\begin{cases} 2 x + 4 y = 8 \\ x + 2 y = 4 \end{cases}$$ If you multiply the second equation by 2, you get the first one. So they represent the **same line** → infinitely many solutions. **Fix**: - If variables disappear and you get a false statement → “No solution”. - If variables disappear and you get a true statement → “Infinitely many solutions”. State this clearly in your final answer. --- ### 6. Careless algebra in linear–quadratic questions When you substitute into a quadratic, one small expansion error can ruin everything. **Fix**: - Write $(x --- > “Practice PSLE Science questions and get clear, step-by-step answers instantly.” > [👉 Try a question now and see how fast you can improve.](https://tutorly.sg/app)  ## Ready to practise? If you want a Singapore-focused AI tutor you can use immediately (website, no sign-up), try Tutorly here: - [https://tutorly.sg/ai-tutor-singapore](https://tutorly.sg/ai-tutor-singapore) - [https://tutorly.sg/app](https://tutorly.sg/app) --- ## Related Articles - ['Online Tutor Help: Smarter, Faster Study Support Singapore' (2026)](/blog/online-tutor-help) - ['Homework Help Online: Expert Guide' (2026): What to do next (2026)](/blog/homework-help-online) - [AI Practice Questions Generator in Singapore: How To Use It Properly (Not Lazily)](/blog/ai-practice-questions-generator-singapore)