Simultaneous EquationsCambridge IGCSE Maths: Revision notes

Simultaneous equations are two (or more) equations that share the same unknowns and must be solved together — the solution is the pair of values that satisfies both equations at once.

• With two linear equations in two unknowns ($x$ and $y$), there is normally exactly one solution
• Simultaneous equations can be constructed from a real-life or geometric situation by translating the given information into two equations
• The two main solving methods are elimination and substitution

Elimination removes one unknown by adding or subtracting the equations.

1. Make the coefficients of one unknown equal (multiply one or both equations)
2. Add the equations if the matching coefficients have opposite signs; subtract if they have the same sign
3. Solve the resulting single-unknown equation
4. Substitute back into either original equation to find the second unknown
5. Check both values in the other original equation

Example: Solve $2x + y = 11$ and $3x - y = 9$. Adding eliminates $y$: $5x = 20$, so $x = 4$. Substituting: $2(4) + y = 11 \Rightarrow y = 3$.

Substitution is often used when one equation is already (or easily made) in the form $y = \ldots$ or $x = \ldots$.

1. Rearrange one equation to make one unknown the subject
2. Substitute that expression into the other equation
3. Solve the resulting single-unknown equation
4. Substitute the value back to find the second unknown

Example: Solve $y = 2x + 1$ and $3x + y = 16$. Substituting: $3x + (2x+1) = 16 \Rightarrow 5x = 15 \Rightarrow x = 3$. Then $y = 2(3)+1 = 7$.

Word problems require translating information into two equations before solving.

1. Choose letters for the unknown quantities
2. Write one equation for each piece of given information
3. Solve using elimination or substitution
4. Check the answer makes sense in context (e.g. quantities should usually be positive)

Example: Two numbers have a sum of 20 and a difference of 4. Let the numbers be $x$ and $y$: $x + y = 20$ and $x - y = 4$. Adding: $2x = 24 \Rightarrow x = 12$, so $y = 8$.

When one equation is linear and the other is quadratic (or otherwise non-linear, with powers no higher than two), use substitution:

1. Rearrange the linear equation to make $x$ or $y$ the subject
2. Substitute into the non-linear equation
3. Solve the resulting quadratic (by factorising, completing the square or the quadratic formula) — this usually gives two solutions
4. Substitute each value back into the linear equation to find the matching second value
5. State both pairs of solutions clearly

Example: Solve $y = x + 1$ and $y = x^2 - 5$. Substituting: $x + 1 = x^2 - 5 \Rightarrow x^2 - x - 6 = 0 \Rightarrow (x-3)(x+2)=0$, so $x = 3$ or $x = -2$, giving $(3, 4)$ and $(-2, -1)$.