Simultaneous Equations

Q: What is the elimination method for simultaneous equations?

The elimination method removes one variable by adding or subtracting the two equations. First, scale the equations so one variable has the same coefficient in both. Then apply SSS (Same Signs Subtract) or DSA (Different Signs Add) to eliminate that variable, solve for the remaining unknown, and substitute back to find the other.

Q: When do you use substitution instead of elimination?

Substitution is used when one equation is already solved for a single variable (e.g. y = 2x + 1), or when one equation is quadratic. Rearrange the simpler (linear) equation, substitute it into the other, and solve the resulting equation.

Q: Are linear-quadratic simultaneous equations on the Higher tier only?

Linear-quadratic simultaneous equations (where one equation is quadratic, such as x squared + y squared = 13) appear only on the Higher tier papers. Standard linear simultaneous equations are tested on both Standard and Higher tiers.

Q: How can a tutor help with simultaneous equations?

A specialist IGCSE maths tutor identifies whether the difficulty lies in setting up equations, choosing a method, algebraic manipulation, or pairing solutions correctly, then provides targeted practice for those specific areas. GetYourTutors provides full-time, in-home tutors across Dubai who specialise in IGCSE Mathematics.

Elimination, substitution and linear-quadratic pairs — the complete IGCSE guide with worked examples for grades 5-9.

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How do you solve simultaneous equations in IGCSE Maths?

For two linear equations, use the elimination method: scale the equations so one variable has matching coefficients, then apply SSS (Same Signs Subtract) or DSA (Different Signs Add) to eliminate it. For linear-quadratic pairs, use substitution: rearrange the linear equation, substitute into the quadratic, solve, and find both pairs of values.

Tier: BothTopic: AlgebraPapers: 1 & 2

Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1

Key Takeaway

Simultaneous equations are high-value questions on both papers. For linear pairs, remember SSS and DSA (Same Signs Subtract, Different Signs Add). For linear-quadratic pairs, always substitute the linear equation into the quadratic — never the other way round. Present answers as coordinate pairs.

How do you solve linear simultaneous equations by elimination?

The elimination method is a five-step process that removes one variable entirely, leaving a single equation with one unknown. Here is the process:

Step 1 — Label the equations: Call them equation (1) and equation (2). This makes your working easy to follow and earns method marks.
Step 2 — Scale the coefficients: Multiply one or both equations so that the coefficient of one variable is the same in both equations.
Step 3 — Eliminate: Look at the signs of the matching coefficients. Same Signs Subtract (SSS) or Different Signs Add (DSA). This removes one variable completely.
Step 4 — Solve: You now have a single equation in one unknown. Solve it.
Step 5 — Substitute back: Put your answer into either original equation to find the second variable.

The mnemonic SSS and DSA is the easiest way to remember whether to add or subtract the equations. If the matching coefficients have the same sign, subtract. If they have different signs, add.

How do you solve linear-quadratic simultaneous equations?

When one equation is linear and the other is quadratic (for example, a circle equation like x² + y² = 13), the substitution method is required. Elimination does not work here because the equations have different degrees.

Step 1 — Rearrange the linear equation to make one variable the subject (e.g. y = 2x + 1).
Step 2 — Substitute this expression into the quadratic equation, replacing that variable everywhere.
Step 3 — Expand and simplify to form a single quadratic equation in one variable.
Step 4 — Solve the quadratic by factorising, completing the square, or using the quadratic formula.
Step 5 — Find both pairs of values: Substitute each solution back into the linear equation to find the corresponding value of the other variable. Always present answers as coordinate pairs.

Worked Examples

Grade 6/7

Solve simultaneously: 5x + 2y = 11 and 3x - y = 11

Label: (1) 5x + 2y = 11 (2) 3x - y = 11

Step 1 (Scale): Multiply (2) by 2: 6x - 2y = 22 ... call this (3)

Step 2 (Eliminate y — different signs, so add): (1) + (3): 11x = 33

Step 3 (Solve): x = 3

Step 4 (Substitute into (2)): 3(3) - y = 11, so 9 - y = 11, y = -2

Check in (1): 5(3) + 2(-2) = 15 - 4 = 11. Correct.

Answer: x = 3, y = -2

Grade 8/9

Solve: y = 2x + 1 and x² + y² = 13

Step 1 (Substitute): Replace y in the quadratic: x² + (2x + 1)² = 13

Step 2 (Expand): x² + 4x² + 4x + 1 = 13

Step 3 (Simplify): 5x² + 4x - 12 = 0

Step 4 (Factorise): (5x - 6)(x + 2) = 0

Step 5 (Solve): x = 1.2 or x = -2

Step 6 (Find y): When x = 1.2: y = 2(1.2) + 1 = 3.4. When x = -2: y = 2(-2) + 1 = -3

Answer: (1.2, 3.4) and (-2, -3)

Grade 9

Solve: x + y = 4 and xy = 3

Step 1 (Rearrange linear): y = 4 - x

Step 2 (Substitute into xy = 3): x(4 - x) = 3

Step 3 (Expand): 4x - x² = 3, so x² - 4x + 3 = 0

Step 4 (Factorise): (x - 3)(x - 1) = 0

Step 5 (Solve): x = 3 or x = 1

Step 6 (Find y): When x = 3: y = 1. When x = 1: y = 3

Answer: (3, 1) and (1, 3)

Common Mistakes to Avoid

The matching pairs trap: When a quadratic gives two x values, students sometimes pair the first x with the second y. Always substitute each x individually into the linear equation to find its corresponding y.

The square expansion trap: (2x + 1)² = 4x² + 4x + 1, not 4x² + 1. Forgetting the middle term when squaring a binomial is one of the most frequent errors on Higher papers.

Substituting into the wrong equation: After solving, always substitute back into the linear equation (not the quadratic). The linear equation is simpler and less prone to arithmetic errors.

Exam Tips for Simultaneous Equations

Show method clearly: Label your equations, show scaling, and write "add" or "subtract" explicitly. Examiners award method marks at every stage.

Give answers as coordinate pairs: For linear-quadratic questions, always present your final answers as (x, y) pairs. Listing x values and y values separately risks losing presentation marks.

Check both values in both equations: Substitute your final pair(s) into both original equations to confirm they satisfy each one. This catches sign and arithmetic errors before you move on.

Expect two solution pairs: Linear-quadratic systems almost always have two solutions. If you only find one, recheck your quadratic — you may have a factorising error.

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How do you solve simultaneous equations in IGCSE Maths?

Tier: BothTopic: AlgebraPapers: 1 & 2

Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1

How do you solve linear simultaneous equations by elimination?

The elimination method is a five-step process that removes one variable entirely, leaving a single equation with one unknown. Here is the process:

Step 1 — Label the equations: Call them equation (1) and equation (2). This makes your working easy to follow and earns method marks.

Step 2 — Scale the coefficients: Multiply one or both equations so that the coefficient of one variable is the same in both equations.

Step 3 — Eliminate: Look at the signs of the matching coefficients. Same Signs Subtract (SSS) or Different Signs Add (DSA). This removes one variable completely.

Step 4 — Solve: You now have a single equation in one unknown. Solve it.

Step 5 — Substitute back: Put your answer into either original equation to find the second variable.

The mnemonic SSS and DSA is the easiest way to remember whether to add or subtract the equations. If the matching coefficients have the same sign, subtract. If they have different signs, add.

How do you solve linear-quadratic simultaneous equations?

Step 1 — Rearrange the linear equation to make one variable the subject (e.g. y = 2x + 1).

Step 2 — Substitute this expression into the quadratic equation, replacing that variable everywhere.

Step 3 — Expand and simplify to form a single quadratic equation in one variable.

Step 4 — Solve the quadratic by factorising, completing the square, or using the quadratic formula.

Step 5 — Find both pairs of values: Substitute each solution back into the linear equation to find the corresponding value of the other variable. Always present answers as coordinate pairs.

Worked Examples

Grade 6/7

Solve simultaneously: 5x + 2y = 11 and 3x - y = 11

Label: (1) 5x + 2y = 11 (2) 3x - y = 11

Step 1 (Scale): Multiply (2) by 2: 6x - 2y = 22 ... call this (3)

Step 2 (Eliminate y — different signs, so add): (1) + (3): 11x = 33

Step 3 (Solve): x = 3

Step 4 (Substitute into (2)): 3(3) - y = 11, so 9 - y = 11, y = -2

Check in (1): 5(3) + 2(-2) = 15 - 4 = 11. Correct.

Answer: x = 3, y = -2

Grade 8/9

Solve: y = 2x + 1 and x² + y² = 13

Step 1 (Substitute): Replace y in the quadratic: x² + (2x + 1)² = 13

Step 2 (Expand): x² + 4x² + 4x + 1 = 13

Step 3 (Simplify): 5x² + 4x - 12 = 0

Step 4 (Factorise): (5x - 6)(x + 2) = 0

Step 5 (Solve): x = 1.2 or x = -2

Step 6 (Find y): When x = 1.2: y = 2(1.2) + 1 = 3.4. When x = -2: y = 2(-2) + 1 = -3

Answer: (1.2, 3.4) and (-2, -3)

Grade 9

Solve: x + y = 4 and xy = 3

Step 1 (Rearrange linear): y = 4 - x

Step 2 (Substitute into xy = 3): x(4 - x) = 3

Step 3 (Expand): 4x - x² = 3, so x² - 4x + 3 = 0

Step 4 (Factorise): (x - 3)(x - 1) = 0

Step 5 (Solve): x = 3 or x = 1

Step 6 (Find y): When x = 3: y = 1. When x = 1: y = 3

Answer: (3, 1) and (1, 3)

Common Mistakes to Avoid

The square expansion trap: (2x + 1)² = 4x² + 4x + 1, not 4x² + 1. Forgetting the middle term when squaring a binomial is one of the most frequent errors on Higher papers.

Substituting into the wrong equation: After solving, always substitute back into the linear equation (not the quadratic). The linear equation is simpler and less prone to arithmetic errors.

Exam Tips for Simultaneous Equations

Show method clearly: Label your equations, show scaling, and write "add" or "subtract" explicitly. Examiners award method marks at every stage.

Give answers as coordinate pairs: For linear-quadratic questions, always present your final answers as (x, y) pairs. Listing x values and y values separately risks losing presentation marks.

Check both values in both equations: Substitute your final pair(s) into both original equations to confirm they satisfy each one. This catches sign and arithmetic errors before you move on.

Expect two solution pairs: Linear-quadratic systems almost always have two solutions. If you only find one, recheck your quadratic — you may have a factorising error.