Systems of linear equations

Solve two linear equations together by substitution or elimination, and recognise when a system has no solution or infinitely many.

Also known as: simultaneous equations, linear systems.

SAT · Algebra GCSE Maths · Algebra IGCSE Maths · Algebra and graphs

Key ideas

A solution is a pair of values that makes both equations true at once.
Elimination: scale equations so one variable cancels when you add or subtract.
Substitution: rearrange one equation for a variable and substitute into the other.
Parallel lines (same gradient, different intercept) mean no solution.

What a solution means

A system (or set of simultaneous equations) is two or more equations that must be true together. For

$x + y = 10, \qquad x - y = 4$

the solution is $x = 7,\ y = 3$ , because both equations hold. Graphically, it is the point where the two lines cross.

Elimination

Add or subtract the equations so one variable disappears. Above, adding gives $2x = 14$ , so $x = 7$ ; substitute to get $y = 3$ .

If no variable cancels immediately, scale first. For $4x - 3y = 18$ and $2x + 5y = -4$ , multiply the second equation by $2$ to get $4x + 10y = -8$ , then subtract the first: $13y = -26$ , so $y = -2$ and $x = 3$ .

Substitution and special cases

When one equation already isolates a variable (e.g. $y = 2x$ ), substitute it: $x + 2x = 12$ gives $x = 4$ , $y = 8$ .

If the two equations describe parallel lines, there is no solution. If they describe the same line, there are infinitely many solutions. Compare the ratios of the coefficients to decide.

Worked examples

Example 1 · Word problem

Tea costs 2 and coffee costs 3 (same currency). A café sold 20 drinks for 49 in total. How many coffees were sold?

Step 1.Let $t$ = teas, $c$ = coffees: $t + c = 20$ and $2t + 3c = 49$ .
Step 2.Substitute $t = 20 - c$ : $2(20 - c) + 3c = 49$ .
Step 3. $40 + c = 49$ , so $c = 9$ .

Answer: 9 coffees

Example 2 · No-solution system

For what value of $a$ does $3x + ay = 7$ and $6x + 8y = 5$ have no solution?

Step 1.No solution means parallel lines: the coefficients must be proportional, $\frac{3}{6} = \frac{a}{8}$ .
Step 2.So $a = 4$ .
Step 3.Check the constants are not in the same ratio: $\frac{7}{5} \neq \frac{1}{2}$ ✓.

Answer: $a = 4$

Common mistakes

Subtracting equations but forgetting to subtract every term, especially negative ones.
Stopping after finding one variable: a solution needs both values.
Scaling only one side of an equation.

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Systems of linear equationsEasier

If $x + y = 10$ and $x - y = 4$ , what is the value of $x$ ?

Question 2Systems of linear equationsEasier

If $y = 2x$ and $x + y = 12$ , what is the value of $y$ ?