Algebra · step 10 of 13

Systems of two equations & substitution

Two linear equations in two unknowns represent two lines, so the system has one solution (crossing lines), none (parallel), or infinitely many (identical lines). Substitution suits systems where one variable is already isolated; elimination suits matching coefficients. Non-linear systems — a line and a circle, say — are handled by substitution and can have two solutions.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Choose substitution or eliminationSubstitution when a variable has coefficient ±1; elimination when coefficients can be matched by small multipliers.
Substitute and solve one variableExpress one unknown in terms of the other and substitute into the second equation, giving a single-variable equation.
Back-substitute for the second variablePut the value found into the simpler original equation.
Check in BOTH original equationsA solution satisfying only one equation is not a solution, and the check catches arithmetic slips immediately.

Worked example

Solve y = x + 1 and x² + y² = 25.

Substitute the linear equation into the circle: x² + (x + 1)² = 25.
Expand: x² + x² + 2x + 1 = 25, so 2x² + 2x − 24 = 0, i.e. x² + x − 12 = 0.
Factor: (x + 4)(x − 3) = 0, giving x = 3 or x = −4.
Back-substitute into y = x + 1: x = 3 gives y = 4; x = −4 gives y = −3.

Answer. Two intersection points: (3, 4) and (−4, −3) — both satisfy 9 + 16 = 25.

Where marks get dropped

These are the specific errors that cost credit on systems of two equations & substitution questions — QED's rubric penalises each of them separately.

Reporting only one solution for a line-and-circle system. A line usually cuts a circle twice.
Substituting back into the equation you just used to eliminate, which is automatically satisfied and proves nothing.
Concluding "no solution" from 0 = 0. That signals infinitely many solutions; 0 = 5 is the no-solution case.

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Systems of two equations & substitution — frequently asked questions

When does a linear system have no solution?

When the lines are parallel but distinct — the same coefficient ratio with a different constant. Algebraically you reach a contradiction like 0 = 5.

How do I recognise infinitely many solutions?

One equation is a multiple of the other, and elimination gives 0 = 0. The solution set is the whole line, parametrised by one variable.

Is elimination better than substitution?

For larger systems yes — it generalises to Gaussian elimination. For two equations with an isolated variable, substitution is usually quicker.

The rest of Algebra

Foundational algebra to close prerequisite gaps. Each subtopic below has its own method, worked example and mark-losing traps.

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