Year 12 Algebra: Simultaneous equations

Simultaneous-equation questions ask you to satisfy more than one algebraic condition at the same time. In Year 12, that can mean linear systems with several unknowns or systems where one equation is nonlinear. The common theme is organisation. You are not just solving one equation and then another. You are coordinating them so that every final answer works in the whole system. This is why checking is essential. A value that fits one transformed equation may still fail in the original pair or set.

Eliminate cleanly in linear systems, substitute strategically in nonlinear systems, and always check the final solution in the original equations.

Subtopic 1: Linear systems with several unknowns

For linear systems, the aim is to reduce the number of unknowns step by step. In a three- or four-variable system, that usually means combining equations to eliminate one variable, then repeating the process on the smaller system that remains. A neat layout matters because it is very easy to lose a sign or mismatch coefficients once several equations are in play.

Worked example 1

Problem: Solve

x + y + z = 9

x + y - z = 5

, and

x - y + z = 3

Subtract the second equation from the first: $2 z = 4$ , so $z = 2$ .
Substitute $z = 2$ into the second equation: $x + y - 2 = 5$ , so $x + y = 7$ .
Substitute $z = 2$ into the third equation: $x - y + 2 = 3$ , so $x - y = 1$ .
Add the last two equations: $2 x = 8$ , so $x = 4$ .
Then $y = 3$ .

Answer:

x = 4

y = 3

z = 2

Worked example 2

Problem: Solve the four-variable system

x + y + z + w = 10

x + y + z - w = 6

x + y - z + w = 8

, and

x - y + z + w = 4

Subtract the second equation from the first: $2 w = 4$ , so $w = 2$ .
Subtract the third equation from the first: $2 z = 2$ , so $z = 1$ .
Subtract the fourth equation from the first: $2 y = 6$ , so $y = 3$ .
Substitute into the first equation: $x + 3 + 1 + 2 = 10$ .
So $x = 4$ .

Answer:

x = 4

y = 3

z = 1

w = 2

Subtopic 2: Simultaneous nonlinear equations

Nonlinear systems usually require substitution or comparison of two different forms. A line and a parabola, for example, can create a quadratic after substitution. Once that happens, you must solve the resulting equation and then check each candidate pair back in the original system. This checking step is not optional, because transformed equations can sometimes introduce values that do not truly satisfy the original relationships.

Worked example 3

Problem: Solve

y = x + 1

and

x^{2} + y = 7

Substitute $y = x + 1$ into the second equation: $x^{2} + x + 1 = 7$ .
Rearrange: $x^{2} + x - 6 = 0$ .
Factor: $(x + 3) (x - 2) = 0$ .
So $x = - 3$ or $x = 2$ .
Substitute back into $y = x + 1$ .
This gives $(- 3, - 2)$ and $(2, 3)$ .

Answer: the solutions are

(- 3, - 2)

and

(2, 3)

Why structure matters

Simultaneous-equation problems are really about reducing complexity without losing meaning. In linear systems, elimination reduces the number of unknowns while preserving the system. In nonlinear systems, substitution converts the pair into a single equation you already know how to solve. Once you see that, the topic becomes less mysterious. The challenge is not inventing a new method for every question. It is choosing the right reduction and carrying it out without disrupting the logic of the system.

Common traps

Losing a sign during elimination and then building the rest of the solution on a false equation.
Checking candidate values in a transformed equation only, instead of in the original system.
Substituting into the messier equation when a cleaner substitution path exists.
Stopping after solving for some variables without finishing the whole system.

Revision focus

In revision, label the purpose of each step: eliminate $z$ , substitute for $y$ , or check in the original equation. This keeps long simultaneous work readable and makes it much easier to diagnose errors. It also helps to rewrite each reduced equation neatly on its own line instead of trying to manage everything in one crowded block.

A second useful habit is to compare algebra with geometry when possible. A nonlinear system often represents two graphs meeting. If the algebra gives two solutions, that should fit the geometric story. When the algebra and the geometry agree, confidence in the answer rises sharply.