Equations as scales
About this app
Equations as beam balances: add two scales and knock out an unknown with Gauss – coloured stones against metal weights.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
An equation is a scale
Unknowns lie on the left, numbers on the right, and both sides weigh the same. In the app every unknown is a coloured stone with a hidden weight, the right side consists of metal weights. Scale 1 holds x₁ + x₂ + x₃ = 8, scale 2 holds 2x₁ + 3x₂ + 4x₃ = 28. Put everything from both left pans onto the left pan of a third scale and everything from the right pans onto its right pan, and it stays balanced: 3x₁ + 4x₂ + 5x₃ = 36. That is why you may add equations.
What the app shows
In the scene Add the stones glide onto the big scale in a cascade, it tips, then the weights follow and the scale settles. The wrong attempt "left only" leaves the weights where they are: the scale clearly tips. In the scene Eliminate scale 1 is taken twice and its contents are removed from scale 2. Scale 2 stays balanced and both x₁ stones are gone: x₂ + 2x₃ = 12. Under every scale its equation is shown, with <, = or > depending on the tilt.
Try it yourself
Step forward and back, play everything or pick a step. When eliminating you choose the factor 1, 2 or 3 and see whether an unknown disappears. With 3, x₂ drops out and a missing x₁ ends up on the right side: −x₁ + x₃ = 4. The values panel shows the weights of both pans, the tilt and the equation of the big scale in the current step.
Frequently asked questions
- Why may you add two equations?
- Both sides of an equation are equal. If you add the same on the left as on the right, namely the two equal sides of the second equation, the sides stay equal. Every solution of both equations also satisfies the sum.
- What is the trick in Gaussian elimination?
- You subtract a suitable multiple of one equation from another so that one unknown gets the coefficient 0. 2x₁ + 3x₂ + 4x₃ = 28 minus 2 · (x₁ + x₂ + x₃ = 8) gives x₂ + 2x₃ = 12.
- What happens if you change only one side?
- Then the equation is no longer true. On the scale you see it at once: it tips. Every transformation must do the same to both sides.
- What can I set in the app?
- The scene Add or Eliminate, the steps 0 to 3, the factor 1, 2 or 3 for eliminating, the wrong attempt "left side only" and whether each scale shows its equation.
Subject: Linear algebra
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