havo vwo
Systems of equations
Quick answer
Two lines: a₁x + b₁y = c₁ and a₂x + b₂y = c₂. If D = a₁b₂ − a₂b₁ is not 0, there is one intersection.
Example: x + y = 5 and x − y = 1 gives x = 3, y = 2. When D = 0: no solution, or infinitely many.
Calculator
Enter the six coefficients. The tool uses elimination (the same arithmetic as Cramer’s rule).
What is a system?
A stelsel (Dutch for system of equations) is a set of equations that must hold at the same time. Two linear equations in x and y are two lines in the plane. They intersect, run parallel, or coincide.
| Method | Idea | Handy when… |
|---|---|---|
| Substitution | Isolate one letter, plug it into the other equation | One letter is already almost alone (for example y = …) |
| Elimination | Make one letter match, then add or subtract | Coefficients are easy to match |
Both methods give the same pair (x, y). This tool uses elimination. One equation with one unknown: linear equation solver.
Examples by level
Dutch secondary tracks: havo (middle) and vwo (university-preparatory). Systems of two linear equations are not a fixed exam topic on vmbo-bb/kb in this form, so there is no vmbo tab.
x + y = 5 and x − y = 1 (elimination)
- Add2x = 6 → x = 3
- Back-substitute3 + y = 5 → y = 2
Check: 3 + 2 = 5 and 3 − 2 = 1. Substitution: from the second, y = x − 1; plug into the first: x + (x − 1) = 5 → x = 3, y = 2.
x + y = 2 and 2x + 2y = 5. Solutions?
- D1·2 − 2·1 = 0
- CheckThe second line is not a multiple of the first (right-hand side 5 ≠ 4)
- ConclusionParallel: no solution
With 2x + 2y = 4 the lines would coincide: infinitely many solutions.
Common mistakes
Wrong
Finding only x and forgetting y.
Right
A system asks for a pair: x and y. Check both equations.
Wrong
In elimination, multiplying the whole equation but forgetting the right-hand side.
Right
Left and right by the same factor. Otherwise the balance breaks.
Wrong
Treating D = 0 as “you mistyped”.
Right
D = 0 is a valid case: no solution, or infinitely many.
Practice
havo x + y = 5, x − y = 1. What is (x, y)?
(3, 2)
Add: 2x = 6 → x = 3, y = 2.
vwo x + y = 2 and 2x + 2y = 5.
No solution
D = 0 and the second is not a multiple of the first (5 ≠ 4).
vwo x + y = 2 and 2x + 2y = 4.
Infinitely many solutions — same line.
On havo and vwo
Systems of two linear equations belong on havo and vwo. One equation: linear equation solver. Quadratic in one unknown: quadratic formula calculator.
Next step
- How does this work? The explanation and examples are further up this page.
- See it visually Linear function simulation: change the values and watch what happens.
- Test yourself Linear equation practice, with an explanation after every answer.
- Video explanations On the Linear function page. YouTube only loads when you press play.
Learning path
- Learn first: Linear equation solver Solve equations of the form ax + b = c.
- Related: Quadratic formula calculator
Frequently asked questions
How do you solve a system of two equations?
Substitution or elimination. One intersection point, parallel lines, or the same line.
What is the difference between substitution and elimination?
Substitution: isolate one letter and plug it in. Elimination: make one letter match, then add or subtract. Same solution.
What does no solution mean?
The equations contradict each other. Graphically: parallel lines.
When are there infinitely many solutions?
When both equations describe the same line (multiples of each other, including the right-hand side).