Algebra foundations · 02 · The balance · 8 min
Solving equations
An equation is a claim: these two things are equal. Solving it means finding the value that makes the claim true — and the method is a single rule applied patiently: keep the scale balanced.
Build the intuition
The golden rule
Picture 2x + 3 = 11 as a balanced scale. Whatever you do to one side, do to the other, and balance survives. Subtract 3 from both pans: 2x = 8. Halve both pans: x = 4. The unknown stands alone, and the claim is solved.
Undo, in reverse order
The expression 2x + 3 was built by multiplying then adding. To unwrap it, undo in reverse: subtract first, divide second — like taking off shoes before socks would fail, but socks-then-shoes backwards works. Solving is unwrapping.
Check is free
Substitute your answer back: 2(4) + 3 = 11 ✓. Twenty seconds, total certainty. People who are “good at algebra” are mostly people who always check.
See it move
Slide x and feel the equation: the beam tips while the sides disagree and settles exactly when the claim becomes true.
A worked example
Solve a real one
Your phone plan costs $20 plus $0.10 per text. The bill is $26. How many texts?
Set up the claim:
Subtract 20 from both sides:
Divide both sides by 0.1:
Check: 0.1(60) + 20 = 26 ✓ — sixty texts.
Out in the world
Dosage math in hospitals
Nurses solve equations on every shift: a drug comes as 250 mg per 5 ml, the patient needs 175 mg — how many ml? Setting up and solving the proportion correctly is literally a safety-critical algebra problem.
Common confusion, cleared
“Move the 3 over and it becomes −3 — it's a magic trick.”
No magic: you subtracted 3 from both sides, and on the left it cancelled. The “move it over” shortcut is balance-keeping with the middle step hidden.
“Equations have one solving recipe to memorize.”
There's one principle — keep balance, undo in reverse — and it adapts to every equation. Understand the principle and you never need the recipe card.
Check yourself
PracticeQuick check
Solve: 3x − 5 = 16
To solve 4(x + 2) = 20, the gentlest first move is…
Recap
- An equation is a claim of equality; solving finds what makes it true.
- Do the same thing to both sides — always.
- Undo operations in reverse order, then check by substituting back.