How to Solve Simultaneous Equations for IGCSE (CIE) Maths

How to Solve Simultaneous Equations for IGCSE (CIE) Maths

You get a simultaneous equations question. You know all three methods, elimination, substitution, graphical, so you pick one and start grinding through it. Ten minutes and half a page of algebra later, you're stuck with fractions everywhere and a horrible feeling that this isn't the route the question wanted. That's usually not a knowledge problem. It's that you reached for elimination on a pair built for substitution, or the other way round.

After marking hundreds of scripts on this exact topic, the pattern is always the same. Students who lose marks here almost always know a method. They just picked the wrong one for what was actually in front of them.

This sits inside the Algebra chapter of the CIE IGCSE Extended syllabus (0580), right next to quadratic equations and sequences. If you haven't locked down the four ways to solve a quadratic yet, our guide to solving quadratic equations is worth doing first, since the linear-quadratic method further down leans directly on it.

Quick checklist before you start solving:

• Can you tell elimination, substitution, and the linear-quadratic case apart just by looking at the equations, before you touch a pencil?

• Coefficients of x or y already matching, or easy to scale so they match? That's elimination.

• One equation already reads "y = " or "x = "? That's substitution.

• One equation has an x² or y² in it? Substitution only, straight into the quadratic.

• Checked your final answer by substituting into the equation you didn't rely on last?

What Is a Simultaneous Equation?

A simultaneous equation is really a pair, or a set, of equations that has to be solved together rather than one at a time. The solution is the one set of values for the unknowns that makes every equation true at once. For two linear equations in x and y, that's the single point (x, y) where their graphs cross.

Get one equation right and the other wrong, and you haven't solved the system at all. Even if both answers look sensible sitting on their own.

What Are the Methods for Solving Simultaneous Equations at IGCSE?

CIE IGCSE Extended (0580) expects three methods for two linear equations: elimination, substitution, and the graphical method. There's a fourth case worth knowing too, substitution again, this time straight into a quadratic, for when one equation is linear and the other is quadratic. Exam questions rarely tell you outright which method to use, so spotting the right one just from how the equations are written is worth marks on its own.

Method

Best used when

Pros

Cons

Elimination

Both equations are linear and a variable's coefficients match (or scale to match) easily

Fast, avoids fractions if coefficients line up

Needs careful sign handling when subtracting negatives

Substitution

One equation is already (or easily made) "y = " or "x = "

Always works for linear pairs; the only method that also works when one equation is quadratic

More algebra steps; easy to substitute into the wrong equation

Graphical

The question gives axes, a table of values, or asks for a sketch

Shows the solution visually, and shows how many solutions exist

Reading off exact non-integer values is imprecise

How Do You Solve Simultaneous Equations by Elimination?

Elimination means adding or subtracting the two equations so that one variable cancels out completely, leaving a single equation in one unknown you can solve directly. It works best when the coefficients of x or y already match, or can be scaled to match by multiplying one or both equations.

Worked example: Solve 3x + 2y = 12 and 2x - 2y = 4.

Step 1: Look at the y-coefficients. They're +2 and -2, so adding the two equations cancels y straight away.

(3x + 2y) + (2x - 2y) = 12 + 4 5x = 16 x = 3.2

Step 2: Substitute x = 3.2 back into either original equation to find y. Using 2x - 2y = 4:

2(3.2) - 2y = 4 6.4 - 2y = 4 -2y = -2.4 y = 1.2

Step 3: Check both original equations with (x, y) = (3.2, 1.2). Always do this on an exam script. A checked answer is a marked answer, even when the working before it slipped somewhere.

How Do You Solve Simultaneous Equations by Substitution?

Substitution means rearranging one equation to make x or y the subject, then substituting that expression into the other equation so only one unknown remains. Reach for it whenever one equation already starts with "y =" or "x =". It's also the only method that extends to the linear-quadratic case below.

Worked example: Solve x + y = 10 and 2x - y = 4.

Step 1: Rearrange the first equation to make x the subject.

x = 10 - y

Step 2: Substitute into the second equation.

2(10 - y) - y = 4 20 - 2y - y = 4 20 - 3y = 4 3y = 16 y = 16/3

Step 3: Substitute back to find x.

x = 10 - 16/3 = 14/3

That answer isn't a tidy integer, and that's deliberate. CIE Extended papers mix clean and messy solutions on purpose, so a method that only works for "nice" numbers isn't reliable exam technique.

How Do You Solve a Linear and a Quadratic Equation Simultaneously?

Substitute the linear equation into the quadratic equation to eliminate one variable. That leaves a quadratic in a single unknown: solve it by factorisation or the formula to get up to two x-values, then substitute each one back into the linear equation to find its matching y-value. This case only appears at Extended tier, and it's the one place elimination never applies, because you can't cancel a squared term against a linear one.

Worked example: Solve the simultaneous equations y = 3x - 2 and y = x² - 2.

Step 1: Since both are already "y = ", set the right-hand sides equal to each other.

x² - 2 = 3x - 2

Step 2: Rearrange into the standard quadratic form ax² + bx + c = 0.

x² - 3x = 0

Step 3: Factorise and solve for x.

x(x - 3) = 0 x = 0 or x = 3

Step 4: Substitute each x-value into the linear equation, never the quadratic, since that's more error-prone, to find y.

x = 0: y = 3(0) - 2 = -2 x = 3: y = 3(3) - 2 = 7

Final answer: (x, y) = (0, -2) or (3, 7).

Two solutions is the norm here, because a straight line crossing a parabola usually does so at two points. One solution means the line is a tangent to the curve. Zero real solutions means the line misses the curve entirely, and the discriminant of the resulting quadratic will tell you which before you even factorise.

What Is the Graphical Method for Simultaneous Equations?

The graphical method means plotting both equations on the same axes and reading off the coordinates of every point where they cross. Each intersection point is one solution to the system. For two linear equations, one crossing point means one solution, parallel lines mean no solution, and identical lines mean infinitely many solutions. For a linear-quadratic pair, the number of crossing points (0, 1, or 2) tells you the number of solutions before you calculate anything at all, which is a fast way to sanity-check algebraic working.

What Mistakes Cost Marks on This Topic?

• Picking elimination for a pair that's really a substitution setup (or the other way round), and grinding through unnecessary algebra as a result.

• Sign errors when subtracting a negative during elimination, especially when one coefficient is already negative.

• Substituting an x-value back into the quadratic equation instead of the linear one in the linear-quadratic case, which is both slower and more error-prone.

• Skipping the check step, then running out of time to catch a sign slip that happened three lines earlier.

• Reading a graphical solution to more decimal places than the graph can actually support.

Frequently Asked Questions

What's the difference between elimination and substitution? Elimination adds or subtracts two equations to cancel a variable directly. Substitution replaces a variable in one equation with an expression from the other. Both give the same answer for a linear pair. The "right" one is whichever needs fewer steps for the specific equations in front of you, and substitution is the only one of the two that also works when one equation is quadratic.

Can simultaneous equations have no solution? Yes. Two linear equations with no solution represent parallel lines that never meet. Algebraically, you'll end up eliminating both variables and being left with a false statement, something like 0 = 6. For a linear-quadratic pair, no real solution means the line doesn't touch the curve, which shows up as a negative discriminant in the resulting quadratic.

Do I always need to check my answer? Yes, and on CIE Extended papers it costs seconds but recovers marks. Substitute both values you found back into the equation you didn't use to solve for them last. If it doesn't balance, a sign error happened earlier, and you can often find and fix it before moving on.

Why does the linear-quadratic method always substitute the linear equation into the quadratic? Substituting the linear expression for y (or x) into the quadratic equation keeps the algebra to one variable and one squared term, which you can then solve with familiar quadratic methods. Substituting the quadratic into the linear equation the other way round doesn't remove the squared term and leaves you no further forward.

Is the graphical method accurate enough for exam marks? Only when the solutions are whole numbers, or the question explicitly asks for a graphical or estimated answer. CIE mark schemes for graphical questions usually accept a tolerance range. For any question asking you to "solve" or "find the exact values," use elimination or substitution instead, since reading a graph can't reliably hand you a fraction like 16/3.

How Should You Revise This Topic?

Don't drill one method in isolation and assume the skill transfers. Take ten past-paper simultaneous equations questions, cover the mark scheme, and spend the first thirty seconds on each one just deciding which method fits, before you write a single line of working. That decision is the actual skill this topic tests.

If you want a set of past paper questions filtered specifically to this topic, the Question Bank has them broken down by method, so you can drill elimination, substitution, and the linear-quadratic case separately instead of getting them jumbled together. And if you've worked through a full solution and aren't sure whether your method choice or your working would actually pick up full marks, the AI Tutor can mark it against the same criteria examiners use.