In short

An equation like 3(x+2) - 5 = 7 is a tower of operations sitting on top of x. Reading it from x outwards, the operations are: add 2, multiply by 3, subtract 5. To solve the equation you UNDO those operations — but in REVERSE order, like unwrapping a Diwali gift. The outermost wrapping (the -5) comes off first by adding 5. Then the multiplication by 3 comes off by dividing by 3. The innermost addition of 2 comes off last by subtracting 2. Last on, first off — that is solving.

Hand a small child a beautifully wrapped Diwali present and watch what they do. They tear off the outer layer of paper first. Then the next layer. Only at the end, when every wrapping is on the floor, do they reach the toy inside. Nobody ever tries to tear the toy out from underneath — the wrappings are in the way, and they came on in a particular order, so they have to come off in the opposite order.

Solving an equation works exactly the same way. The variable x is the toy. The operations done to x — adding, multiplying, dividing, subtracting — are the layers of wrapping. Your job is to unwrap them, one by one, until x stands alone. And the rule for the order is non-negotiable: the layer that went on last comes off first.

This article gives you a single picture for solving any linear equation, no matter how tangled it looks. Forget memorising "do this then that." Once you see an equation as a stack of operations on x, the solving steps write themselves.

The wrap view of an equation

Take the equation 3(x+2) - 5 = 7.

Read the left side from x outwards. What is the very first thing that happens to x? You add 2 to it, getting x+2. What happens next? You multiply by 3, getting 3(x+2). What happens after that? You subtract 5, getting 3(x+2) - 5. So the build order is:

  1. Start with x.
  2. Add 2.
  3. Multiply by 3.
  4. Subtract 5.

That is the wrap order — the order in which operations were piled on top of x. Picture them as a stack of boxes, each one nesting around the previous, with x at the very centre.

To get x back out, you have to unwrap. And you cannot reach into the centre of a stack of boxes — you must take the outermost one off first. So the unwrap order is the wrap order in reverse:

  1. Undo the subtract 5 (outermost) — add 5 to both sides.
  2. Undo the multiply by 3 — divide both sides by 3.
  3. Undo the add 2 (innermost) — subtract 2 from both sides.

Apply that to 3(x+2) - 5 = 7:

3(x+2) - 5 = 7

Add 5 to both sides:

3(x+2) = 12

Divide both sides by 3:

x + 2 = 4

Subtract 2 from both sides:

x = 2

Three steps. Each one peels off the outermost surviving operation. The work is mechanical — no cleverness needed once the stack is in your head.

Why the order MUST be reverse: the operations form a stack — Last In, First Out. The subtract 5 is on the outside of the multiplication by 3, so the equation literally reads "(three times something) minus 5 equals 7." The "minus 5" is the easiest to see and the easiest to undo, because nothing else is wrapped around it. If you tried to "undo the +2" first by subtracting 2, you would be trying to subtract 2 from a quantity (3(x+2) - 5) that does not contain a bare +2 at all — you would have to break through the multiplication and the subtraction first to reach it. Function inverses compose in reverse order: if f(x) = 3(x+2) - 5 is built by chaining "add 2," "multiply by 3," "subtract 5," then f^{-1} chains the inverses in the opposite order. This is the same rule that makes you take off your shoes before your socks.

The unwrap widget

Below is a stack of operation cards sitting on top of x — top of the stack is the outermost operation. Click Pop top to undo the outermost operation. Watch the equation shrink one layer at a time.

2(x + 4) − 3 = 9
Operations on x (top = outermost)
Steps applied (so far)

What the widget shows, in plain words: the stack has three cards. The top is subtract 3 because that is the very last thing done to x + 4 on the left side of 2(x+4) - 3 = 9. Pop it, and the equation becomes 2(x+4) = 12 — the subtraction is gone, undone by adding 3 to both sides. The new top of the stack is multiply by 2. Pop it, and you get x + 4 = 6. The last card is add 4, the innermost wrapping. Pop it, and you are left with x = 2.

The Diwali gift picture

Equation as a Diwali gift with three layers of wrappingA diagram of a present with three concentric square layers around a central x. The outermost layer is labelled subtract 3, the middle layer multiply by 2, the innermost layer add 4. Arrows on the right show the unwrapping order from outermost to innermost. outer layer: subtract 3 middle layer: multiply by 2 inner layer: add 4 x 1st: peel off (add 3) 2nd: peel off (÷ 2) 3rd: peel off (sub 4) Last on, first off. The toy ($x$) is reached only when every layer is gone.
The equation $2(x+4) - 3 = 9$ as a Diwali gift. Three layers of wrapping sit around the central $x$. To find $x$, you peel the outer wrapping first (subtract $3$ → add $3$), then the middle (multiply by $2$ → divide by $2$), then the inner (add $4$ → subtract $4$). The order is the reverse of how the wrappings were put on.

Worked examples

$5x + 2 = 17$ — two layers

The wrap view. Starting from x:

  1. Multiply by 5 → gives 5x.
  2. Add 2 → gives 5x + 2.

Stack (top is outermost): add 2, then multiply by 5.

The unwrap. Pop the top first.

5x + 2 = 17 \quad\xrightarrow{\text{subtract } 2}\quad 5x = 15 \quad\xrightarrow{\text{divide by } 5}\quad x = 3.

Check: 5(3) + 2 = 15 + 2 = 17. Correct.

Why subtract before divide: the multiplication by 5 is wrapped around the variable, and the addition of 2 is wrapped around the multiplication. So the +2 is the outermost layer. If you divided by 5 first, you would be dividing 5x + 2 by 5, which gives x + \tfrac{2}{5} — fractions appear that did not need to. Peeling the outer layer first keeps the arithmetic clean.

$\dfrac{x - 3}{4} = 5$ — two layers, one is a division

The wrap view. Starting from x:

  1. Subtract 3 → gives x - 3.
  2. Divide by 4 → gives \dfrac{x-3}{4}.

Stack: divide by 4 (outermost), then subtract 3 (innermost).

The unwrap.

\frac{x-3}{4} = 5 \quad\xrightarrow{\text{multiply by } 4}\quad x - 3 = 20 \quad\xrightarrow{\text{add } 3}\quad x = 23.

Check: \dfrac{23 - 3}{4} = \dfrac{20}{4} = 5. Correct.

The trap of skipping order — what goes wrong

Take 5x + 2 = 17 again, but try to undo the inner layer first. "I will subtract 5x to get rid of the 5x term," you say. That gives 2 = 17 - 5x. Now you are stuck with x on the right, and the equation is no simpler than before — it just looks different.

Or worse, on \dfrac{x-3}{4} = 5, suppose you try to "undo the -3 first" by adding 3 to both sides. You get \dfrac{x-3}{4} + 3 = 8. The -3 is still trapped inside the fraction — adding 3 on the outside did not touch it. The new equation is uglier than the original, and you have learned nothing.

The lesson: you cannot reach the inner layer through the outer one. The outer layer has to come off first, every time. Why: each operation has a region of influence. The outer "-3" in 5x + 2 = 17 — wait, that is the wrong example, the outer was +2. The outer "+2" acts on the entire 5x block. To cancel it, you must add or subtract on both sides — and that operation must be the first thing you do, before you start picking apart 5x itself. Once +2 is gone, the inner block 5x stands alone, and now you can divide. Inverse order is not a stylistic choice; it is the only order that works without producing a worse equation.

Why this is the universal CBSE Class 7-8 algorithm

Every solving technique you learn in CBSE Class 7-8 algebra — "transposition," "do the same to both sides," "balance method" — is the same idea wearing a different uniform. Each one identifies the outermost operation on x and undoes it. Transposition is just the shorthand: "+2 on the left becomes -2 on the right" because to remove +2 from the left, you subtract 2 from both sides, and the right side becomes "the old right minus 2." The reasoning is unwrap-from-the-outside.

Once the stack picture is in your head, you do not need to memorise rules at all. You look at the left side of the equation, ask "what is the very last thing being done to x?" — that is the top of the stack. Undo it. Repeat. When the stack is empty, x is alone on one side, and the right side is the answer.

This generalises far beyond linear equations. In trigonometry, \sin(2x + 30°) = \tfrac{1}{2} is solved by peeling: outermost is \sin, undo with \sin^{-1}; then "+30°", undo with -30°; then "multiply by 2", undo with \div 2. In logarithms, \log_2(3x - 1) = 4 is solved by peeling: outermost is \log_2, undo by raising 2 to the power. The unwrapping principle does not change.

The takeaway

Three rules, no exceptions:

  1. Read the left side from x outwards to find what was done, in order.
  2. The last operation done is the first one to undo. Top of the stack pops first.
  3. Apply each undo to BOTH sides so the equation stays balanced.

When you sit down with 7(2x - 5) + 3 = 24, do not panic and do not search for "the trick." Identify the stack: starting from x, you multiplied by 2, subtracted 5, multiplied by 7, added 3. The top is +3. Subtract 3. Then divide by 7. Then add 5. Then divide by 2. You will get x = \tfrac{38}{14} = \tfrac{19}{7}, and you will have done it without thinking, because the stack told you what to do.

Solving an equation is not a puzzle. It is unwrapping a present.

References

  1. NCERT Class 7 Mathematics, Chapter 4: Simple Equations — official CBSE introduction to balance-method solving.
  2. NCERT Class 8 Mathematics, Chapter 2: Linear Equations in One Variable — extends the same algorithm to brackets and fractions.
  3. Khan Academy: Why we do the same thing to both sides of an equation — companion intuition for the balance principle.
  4. Paul's Online Math Notes: Solving Linear Equations — concise reference written for self-study.
  5. Wikipedia: Inverse function — the formal reason inverses compose in reverse order, for readers ready for the abstract version.