Systems of Equations

A system of equations is a set of two or more equations with the same variables, solved simultaneously.

What is a System of Equations?

A system of linear equations consists of two or more equations with the same unknowns:

$\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$

A solution is an ordered pair $(x, y)$ that satisfies all equations simultaneously.

Methods of Solution

Substitution Method

1
Solve one equation for one variable
Choose the simpler equation and isolate one variable.
2
Substitute into the other equation
Replace the isolated variable in the second equation.
3
Solve the resulting equation
This gives the value of one variable.
4
Back-substitute to find the other variable
Substitute the found value into the expression from Step 1.

Elimination Method

1
Multiply equations to match coefficients
Multiply one or both equations so that one variable has equal (or opposite) coefficients.
2
Add or subtract equations
Eliminate one variable by adding or subtracting the equations.
3
Solve for the remaining variable
The result is a one-variable equation.
4
Find the second variable
Substitute back to find the other unknown.

System: x + y = 5, 2x − y = 4

Using elimination:

Add the equations: $3x = 9 \implies x = 3$

Substitute: $3 + y = 5 \implies y = 2$

Solution: $(3, 2)$

Types of Solutions

Type	Meaning	Geometric Interpretation
One solution	Lines intersect at one point	Consistent, independent
No solution	Lines are parallel	Inconsistent
Infinitely many	Lines are identical	Consistent, dependent

Note

A system with no solution is called inconsistent. A system with infinitely many solutions is called dependent.

Real-World Applications

Real world: Comparing two shops

Shop A charges €5 per item plus a €3 delivery fee; Shop B charges €3 per item plus a €9 fee. To find the break-even quantity: $5x + 3 = 3x + 9 \implies 2x = 6 \implies x = 3$ items. For more than 3 items, Shop B is cheaper.

Real world: Mixing two drinks

Mix a 20% juice drink and a 60% juice drink to make 400 ml of 35% juice. Let $a$ ml of 20% and $b$ ml of 60% be used: $a + b = 400$ and $0.20a + 0.60b = 140$ . Solving gives $a = 250$ ml and $b = 150$ ml.

Real world: Comparing tradespeople

Electrician A charges €40 call-out plus €20/h; Electrician B charges €10 call-out plus €35/h. They cost the same when $40 + 20h = 10 + 35h \implies 15h = 30 \implies h = 2$ hours. For jobs longer than 2 hours, Electrician A is cheaper.

Tip

Where do you see this? Comparing mobile phone tariffs, mixing solutions in chemistry, finding break-even points in a business, comparing quotes from different contractors.

Practice Problems

Solve: x + y = 7 and x − y = 3

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Add equations: 2x = 10 → x = 5. Substitute: 5 + y = 7 → y = 2. Solution: (5, 2)

Solve by substitution: y = 2x and x + y = 9

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Substitute: x + 2x = 9 → 3x = 9 → x = 3, y = 6. Solution: (3, 6)

How many solutions does the system 2x + 4y = 6 and x + 2y = 3 have?

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The second equation times 2 equals the first — infinitely many solutions (dependent system)