Equations

An equation is a balance scale. The equals sign means that the two sides weigh the same.

      2x + 3   =   11
    ┌────────┐   ┌────┐
    │  ●● ▪▪▪│   │▪×11│
    └────────┘   └────┘
         ⚖

The rule is simple: whatever you do to one side you have to do exactly the same to the other. Then the scale stays in balance.

2x + 3 = 11
2x + 3 − 3 = 11 − 3      ← take 3 away from both sides
2x = 8
2x / 2 = 8 / 2           ← divide both sides by 2
x = 4

Always check. Substitute 4 into the original equation: 2·4 + 3 = 11. It checks out.

That check takes five seconds and catches nearly every arithmetic slip.

The working order when the equation is messier:

Step	Do
1	Multiply the brackets out
2	Collect like terms on each side
3	Move all the x to one side, all the numbers to the other
4	Divide by the number in front of x
5	Check by substitution

An example:

3(x + 2) = 5x − 4
3x + 6   = 5x − 4        (1)
3x + 6 − 3x = 5x − 4 − 3x
6 = 2x − 4               (3)
10 = 2x
x = 5                    (4)

Check: 3(5 + 2) = 21 and 5·5 − 4 = 21. ✓   (5)

Three special cases to recognise:

Equation	Becomes	Means
2x + 3 = 2x + 3	0 = 0	every number is a solution
2x + 3 = 2x + 5	3 = 5	no solution
x² = 9	x = 3 or x = −3	two solutions

Why AI cares: training a model is solving an equation — except with millions of unknowns and without an exact answer key. That is why gradient descent is used: you cannot solve for the answer, you have to feel your way towards it.

Set the equation up first, solve it afterwards. The hard part is nearly always the setting up.

#	Problem
1	A number plus 7 is 22. Which number?
2	Three times a number, minus 5, is 16.
3	A taxi costs 50 kr plus 15 kr/km. The trip cost 155 kr. How far?
4	Anna is twice as old as Bo. Together they are 36. How old are they?
5	You buy n tickets at 120 kr each and pay 840 kr.

The key, with the setting up:

1. x + 7 = 22 → x = 15
2. 3x − 5 = 16 → 3x = 21 → x = 7
3. 50 + 15k = 155 → 15k = 105 → k = 7 km
4. x + 2x = 36 → 3x = 36 → x = 12 (Bo), Anna 24
5. 120n = 840 → n = 7

Number 4 is the one that takes the most thought: you have to choose what x is going to mean. Choose «Bo is x» and it is easy. Choose «Anna is x» and you get x + x/2 = 36, which also works but is messier.

That is general advice: let the variable stand for the smallest or simplest of the things being asked for, and you avoid fractions.