What does solving an equation mean?

It means finding the value of the letter-number that makes the left hand side equal to the right hand side. For 2n + 1 = 99 that value is 49, because 2 × 49 + 1 = 99. This value is the solution of the equation.

Why do we do the same thing to both sides of an equation?

Because the two sides are equal to begin with. Adding, subtracting, multiplying or dividing both by the same number gives two new values that are still equal, just as a balanced scale stays level when equal weights are added to or taken off both pans.

Is moving a term across the same as doing one step on both sides?

Yes. Writing 2y + 7 = 21 as 2y = 21 − 7 is a short way of subtracting 7 from both sides, and writing 2y = 14 as y = 14 ÷ 2 is a short way of dividing both sides by 2. The result is the same either way.

Solving Simple Equations

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What is an equation?

An equation is a statement that two expressions are equal, such as 2n + 1 = 99 or 2z + 4 = 5z − 14. The part before the equals sign is the left hand side (LHS) and the part after it is the right hand side (RHS). The letter stands for an unknown number. Solving the equation means finding the value of the letter that makes LHS and RHS equal; that value is the solution.

The equation 2n + 1 = 99 with its LHS, RHS, letter-number and equals sign labelled — Parts of an equation

Trial and error

One way is to try values. For 2n + 1 = 99, n = 5 gives only 11 and n = 30 gives 61. n = 40 gives 81, closer, and n = 50 gives 101, just too much. Then n = 49 gives exactly 99. No other value works, because each step up in n adds 2 to the LHS. Guessing works here, but for 5x − 4 = 7 the answer is not even a whole number, so we need a method that always works.

Trying n = 5, 10, 30, 40, 50 and 49 in 2n + 1 until the result is 99 — Trial and error

Keep the balance

Think of a pan balance. If both pans hold equal weights, taking the same weight off each pan, or adding the same weight to each, keeps it level. An equation behaves the same way. Start with 15 + 8 = 23: add 10 to both sides and you get 33 = 33; multiply both sides by 2 and you get 46 = 46. Whatever you do to one side, do the same to the other, and the equation stays true. (Only division by zero is not allowed.)

Adding, subtracting, multiplying or dividing both sides of 15 + 8 = 23 by the same number keeps the two sides equal — The same step on both sides

Solving step by step

The plan is to leave the unknown alone on one side. For 5x − 4 = 7, the − 4 is in the way, so add 4 to both sides: 5x = 11. Now the factor 5 is in the way, so divide both sides by 5: x = 11/5, or 2.2. Undo additions and subtractions first, then multiplications and divisions.

Solving 5x − 4 = 7: add 4 to both sides, then divide both sides by 5, then check — Solving 5x − 4 = 7

Always check. Put x = 2.2 back: 5 × 2.2 − 4 = 11 − 4 = 7, which is the RHS, so the answer is right. In the same way, 11y + (−5) = 61 gives 11y = 66 and y = 6, and 6 × 11 − 5 = 61 confirms it.

Unknowns on both sides

In 6y + 7 = 4y + 21, the unknown appears on both sides. Subtract 4y from both sides first: 2y + 7 = 21. Then subtract 7: 2y = 14. Then divide by 2: y = 7. Brackets are opened first: 4(m + 6) − 8 = 2m − 4 becomes 4m + 16 = 2m − 4, so 2m = −20 and m = −10.

Solving 6y + 7 = 4y + 21 in two ways: the same step on both sides, and moving terms across — Two ways to solve

A faster way: moving terms across

Look at what each step does. Subtracting 7 from both sides of 2y + 7 = 21 simply makes the + 7 disappear on the left and appear as − 7 on the right: 2y = 21 − 7. So we can write that step at once. Three rules follow:

A term that is added or subtracted moves across with the opposite sign (its additive inverse).
A factor that multiplies one side moves across as division: 2y = 14 gives y = 14 ÷ 2.
A divisor moves across as multiplication: u ÷ 15 = 6 gives u = 6 × 15 = 90.

Three rules for moving a term to the other side of an equation, with examples — Moving a term across

Mind the mistake

Most errors come from moving a term but keeping its sign, dividing only one term of a side, dividing the wrong way round (8z = 2 gives z = 2/8, not 4), and sign slips with brackets: −2(3 − 4x) is −6 + 8x. An equation can also have no solution: x + 4 = x + 5 can never be true.

Four common mistakes in solving equations, each with the right way — Mind the mistake

Why it matters

These few steps solve every equation in the chapter and many later ones. Use them on stories in word problems on equations, and see how Indian mathematicians solved equations in bijaganita: algebra in ancient India. Answers to the textbook questions are in the NCERT solutions for Part 2 Chapter 7; see all of Part 2 Chapter 7 notes.

How to solve simple equations