Systems of linear equations

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

Two numbers make 10 together. Their difference is 2. Which numbers are they? Here there are two unknowns and two equations.

📄 Open worksheet (PDF, German)

The video

In the video we solve the puzzle with the point of intersection, with the addition method and with the substitution method.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

The most important points

A system of equations (ein Gleichungssystem)

Two equations with two variables: I \(x + y = 10\), II \(x - y = 2\). The solution fits both equations. In the coordinate system it is the point of intersection of the two lines.

The lines intersect at (6|4). — The lines intersect at \((6|4)\).

The addition method (das Additionsverfahren)

Add the two equations: \(y\) disappears. \(2x = 12\), \(x = 6\). Put it into I: \(y = 4\).

I + II: + y and - y cancel out. — I + II: \(+ y\) and \(- y\) cancel out.

The substitution method (das Einsetzungsverfahren)

If one equation already says \(y = …\), put that into the other one: \(y = 2x\) in \(x + y = 9\) gives \(x + 2x = 9\), so \(x = 3\) and \(y = 6\). If you substitute a term with plus or minus, write it in a bracket: \(2 \cdot (x + 1)\).

y on its own: substitute. +y and -y: add. — \(y\) on its own: substitute. \(+y\) and \(-y\): add.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

Show the worksheet solutions

Try it yourself

Task

Solve: I \(x + y = 7\), II \(x - y = 1\)

Show solution

I + II: \(2x = 8\), \(x = 4\). In I: \(4 + y = 7\), \(y = 3\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: lineare Gleichungssysteme (Additionsverfahren, Einsetzungsverfahren, Schnittpunkt). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.