∑Math Atelier
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S1 · Algebra · ALG.EQ.LINEAR.BASIC

Solve basic linear equations

An equation stays true when you perform the same valid operation on both sides.

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Intuition

Understand first, compress later.

An equation is like a balance scale. Whatever you change on the left must also happen on the right.

Switch representations
Equation as a balance3x = 20
both sides weigh the same — the beam stays level
The same move on both sides1
No constant in the pan — nothing to take off.2
Divide both sides by 3: 6.667.Check
x tile — the unknown numberunit token — worth 1ten-block — ten unitscoral with a bar — negativebeam — the equals sign
The beam hangs level because both pans weigh the same — that is exactly what the equals sign claims. Take something off the left and you must take the same off the right, or it tips. Ten units are bundled into one ten-block; if a quantity still will not fit in the pan, a single labelled token carries the number instead. Below, the same step is hung on both pans of a small beam, with the equation it produces set beside it.
Formal view

Isolate the variable using inverse operations. Simplify first, perform the same operation on both sides, and verify by substitution.

Why does it work?

Applying the same operation to both sides preserves the solution set, transforming the equation into a simpler equivalent form.