Mathematical Equations
Find the whole number each letter stands for so every equation in the system holds at once.
What the task asks
You are given a small system of equations written with letters standing in for numbers.
Find the number each letter represents so that every equation in the system holds at the same time.
There is always exactly one set of values that works. Start from whichever equation pins a single letter down, then substitute that value into the others.
Rules
- Every letter stands for a whole number from 1 to 20.
- All the equations in a system must hold simultaneously.
- Exactly one set of values satisfies the system, so no guesswork is needed and no second answer exists.
- `·` means multiply and `:` means divide.
Worked example 1
- A + 3 = B
- B = 9
Solution. The second equation gives B directly: B = 9. Substituting into the first gives A + 3 = 9, so A = 6. Whenever one equation names a letter outright, start there.
Worked example 2
- B = 2 · A
- A + B = 15
Solution. The first equation expresses B in terms of A, so replace B in the second: A + 2 · A = 15, that is 3 · A = 15, giving A = 5. Putting that back into the first equation gives B = 10. When no letter is given outright, substitute until one letter is left.
Timing
The subtest gives you 25 minutes for 20 systems, about 75 seconds each. No notes are allowed, so the substitution has to be done in your head.
How this app presents it
- Enter a number for every letter. All of them must be right for the question to count as correct.
- Low difficulty uses 2 equations and 2 letters, medium 3 and 3, high 4 and 4.
- Medium introduces mixed linear and quadratic systems; High always includes a squared letter or a product of letters. Use integer bounds and candidate testing when full substitution would take too long. A product such as A · A means A squared.
- Generated systems are checked exhaustively over the whole 1–20 range, so a question is only ever shown if it has exactly one solution.
The other Core Module task types
- Figure Sequences Work out the rule governing each figure in a 4 x 4 matrix series, then pick the next two matrices.
- Latin Squares Deduce the letter in the question-mark field of a 5 x 5 grid using only row and column elimination.
This is an unofficial practice tool, not affiliated with or endorsed by g.a.s.t. or the TestDaF-Institut. The description above is written for this project; always check the official preparatory materials for the authoritative instructions.