Solving Equations

Undo what was done, in reverse order, doing the same thing to both sides every time.

Think of a number. Multiply it by 3. Add 4. Tell me the answer — 19.

I can now tell you the number you started with, and I never saw it.

think of a number, and do not tell me what it is think of a number x multiply by 3 3x add 4 3x + 4 you get 19 19 undo run the chain backwards: 19 − 4 = 15, then ÷ 3 = 5

Work the chain backwards. The last thing done was add 4, so undo it first: 19 - 4 = 15.

Before that it was multiplied by 3, so undo that next: 15 ÷ 3 = 5. Your number was 5.

Notice the order. Forwards it was × 3 then +4; backwards it is -4 then ÷ 3 — reversed, not repeated.

Written as algebra that chain is 3x + 4 = 19, and solving it is exactly the unwinding you just did. The equation is a record of what was done; solving is doing it in reverse.

Solving three x plus five equals twenty, one step per line 3x + 5 = 20 the equation 3x = 15 subtract 5 from both sides x = 5 divide both sides by 3 3(5) + 5 = 20 ✓ check in the original

Reverse the order of operations

To build 3x + 5 from x, you multiply by 3, then add 5. To take it apart, go backwards: subtract 5 first, then divide by 3.

3x + 5 = 20 ⇒ 3x = 15 ⇒ x = 5.

Undoing in the wrong order still sometimes works, but it usually creates fractions you did not need.

The order to undo

Clear + and - first. Then clear × and ÷. The opposite of PEMDAS.

Variables on both sides

Collect the variable terms on one side and the numbers on the other.

5x - 2 = 3x + 8. Subtract 3x from both sides: 2x - 2 = 8. Add 2: 2x = 10. Divide: x = 5.

A small tip: move the variable to whichever side keeps the coefficient positive. It prevents sign errors.

Equations with parentheses and fractions

Distribute first: 2(x + 3) = 14 becomes 2x + 6 = 14, so x = 4.

Fractions: multiply every term by the common denominator to clear them. x3 + 12 = 2 times 6 gives 2x + 3 = 12, so x = 4.5.

Multiply every term, including the ones without fractions.

One line, one change

Write each step on its own line and change only one thing per line. It is slower on the page and much faster overall, because you can find your mistake instead of starting over.

Always check

Substitute your answer back into the original equation. Not into a line you wrote — into the original, in case you made an error early.

x = 5 in 5x - 2 = 3x + 8: left side = 23, right side = 23. ✓

No solution, or every solution

Sometimes the variable vanishes. 2x + 3 = 2x + 7 becomes 3 = 7 — false, so there is no solution.

2x + 6 = 2(x + 3) becomes 6 = 6 — always true, so every number works.

These are not mistakes. They are real answers.

Worked examples

Variables both sides

Solve 7x - 4 = 2x + 21.

  1. Subtract 2x from both sides: 5x - 4 = 21 Gather the variables on the left, where the coefficient is bigger.
  2. Add 4 to both sides: 5x = 25 Clear the constant.
  3. Divide by 5: x = 5 Undo the multiplication.
  4. Check: 7(5) - 4 = 31 and 2(5) + 21 = 31 ✓ Both sides agree, so it is right.

x = 5

Clear the fractions

Solve 2x3 - 1 = x2 + 4.

  1. Common denominator of 3 and 2 is 6. Multiply everything by 6 to clear fractions.
  2. 4x - 6 = 3x + 24 6 · 2x3 = 4x, 6 · 1 = 6, 6 · x2 = 3x, 6 · 4 = 24.
  3. Subtract 3x: x - 6 = 24 Variables to one side.
  4. Add 6: x = 30 Check in the original: 603 - 1 = 19 and 302 + 4 = 19 ✓

x = 30

A word problem end to end

A taxi charges $3.50 plus $2.25 per mile. A ride cost $23.75. How many miles?

  1. Let m = miles. Name the unknown.
  2. 3.50 + 2.25m = 23.75 Flat fee plus per-mile charge equals the total.
  3. 2.25m = 20.25 Subtract the flat fee from both sides.
  4. m = 20.25 ÷ 2.25 = 9 Divide by the rate.

9 miles.

Words to know

Inverse operation
The operation that undoes another: + undoes -, × undoes ÷.
Isolate
Get the variable alone on one side.
Two-step equation
One needing two inverse operations, like 3x + 5 = 20.
Variables on both sides
An equation like 5x - 2 = 3x + 8.

Mistakes to watch for

Doing something to one side only. The equals sign is a promise that both sides are the same amount; change one and the promise breaks, so every line after it is about a different problem.

Multiplying only some terms when clearing fractions. Multiplying by 12 must hit every term, including whole numbers that were never fractions. Miss one and the equation quietly changes.

Checking against a rewritten line instead of the original. If you made an error on line two, checking against line three will happily confirm it. Always substitute back into the equation you were given.

Undoing the operations in the original order. If the chain was × 3 then +4, the undo is -4 then ÷ 3. Reversing the order is the whole point, and doing them forwards gives a wrong answer that looks tidy.

"Think of a number" puzzles are ancient. The Rhind Papyrus from about 1650 BC contains one: a quantity and its seventh, added together, make 19.

The Egyptian scribe solved it by guessing 7, seeing that it gave 8 instead of 19, and then scaling the guess. The method is called false position.

It works, it is clever, and it is exactly what algebra was invented to replace.

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