Finding the Unknown

Why This Matters

Have you ever played a guessing game? “I am thinking of a number. I add 3 to it and I get 10. What is my number?”

You can probably feel the answer. The number is 7. But how did your brain do that? You worked backwards. You knew the final answer was 10, you knew 3 was added, so you took the 3 away again.

That little move — finding a hidden number from clues — is one of the most powerful ideas in all of maths. Shopkeepers use it. Engineers use it. Your phone uses it thousands of times a second.

In this chapter we give the hidden number a name, we write the clue as a neat little sentence called an equation, and we learn a clean, sure-fire way to find the number every single time. No more guessing. You will be able to crack puzzles that are far too hard to guess.

The Big Idea

An equation is a maths sentence that says two things are equal — it has an ”=” sign in the middle, with a hidden number (a letter) somewhere in it. Think of it as a weighing scale that is perfectly balanced: the left side weighs exactly the same as the right side. To find the hidden number, you keep the scale balanced — whatever you do to one side, you do the exact same thing to the other side. Step by step, you get the letter alone on one side, and the number sitting opposite it is your answer.

Let’s Break It Down

Before we start, let’s quickly remember the idea of a letter that stands for a number. You met this in the chapter on algebraic expressions.

From a situation to an equation (an unknown letter)

Let’s start with a real puzzle from a weighing scale. The scale below is perfectly level — it balances. On the left pan there is a mystery box of weight x, plus three small 1-unit weights. On the right pan there is a single 10-unit weight.

Figure 7.1 shows this picture.

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Because the scale balances, the left side and the right side must weigh the same. We write that as:

x + 3 = 10

This is our first equation. The letter x is the unknown weight we want to find. We turned a real situation (a balanced scale) into a short maths sentence.

You can do this with words too. Here is how a sentence becomes an equation. Take “a number, doubled, then minus 3, gives 7.” Step by step: the number is x, doubled is 2x, minus 3 is 2x − 3, and that “gives 7” means it equals 7. So:

2x − 3 = 7

The skill of writing the clue as an equation is half the battle. Once you have the equation, the rest is just careful steps.

What an equation is — a balance

Let’s be very clear about what an equation really is. An equation has two parts, joined by an ”=” sign.

• The part on the left of the ”=” is called the Left Hand Side, or LHS for short.
• The part on the right of the ”=” is called the Right Hand Side, or RHS.

In x + 3 = 10, the LHS is x + 3 and the RHS is 10.

The ”=” sign is a promise. It promises that the LHS and the RHS have the same value. That is why the weighing-scale picture fits so well — a balanced scale also says “this side equals that side.”

Solving the equation means finding the value of the letter that makes the promise true — the value that makes LHS exactly equal to RHS. For x + 3 = 10, the answer is x = 7, because 7 + 3 really is 10.

Solving by keeping the balance — do the same to both sides

Here is the heart of the whole chapter. How do we find x?

Look again at the balanced scale for x + 3 = 10. The left pan has the box x plus three 1-weights. We want the box alone on the pan, because then whatever is on the other pan is its weight.

So let’s lift those three 1-weights off the left pan. But wait — if we remove weight from only the left pan, the scale will tip! The left side becomes lighter, and the balance breaks.

The trick: remove the same 3 weights from the right pan too. Then both sides lose exactly 3. The scale stays perfectly level. Figure 7.2 shows this.

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In symbols, that one move looks like this. We subtract 3 from both sides:

x + 3 − 3 = 10 − 3

x = 7

That’s it. The +3 and the −3 on the left cancel out (they undo each other), leaving x alone. On the right, 10 − 3 = 7. So x = 7.

This is the golden rule, and it is the only rule you really need:

Whatever you do to one side of an equation, do the exact same thing to the other side. Then the two sides stay equal — the scale stays balanced. You can add the same number to both sides, subtract the same number, multiply both sides by the same number, or divide both sides by the same number.

Why is this allowed? Because the two sides start out equal. If two amounts are the same, and you change both of them in exactly the same way, they are still the same afterwards. Two equal piles of rice, take a handful off each — still equal. Same idea.

Let’s use the rule on a fresh equation, with the answer revealed only at the end.

Notice the pattern. To get x alone, you undo whatever is being done to it, using the opposite operation — and you do that undoing to both sides.

• To undo + a number, you subtract that number.
• To undo − a number, you add that number.
• To undo × a number, you divide by that number.
• To undo ÷ a number, you multiply by that number.

Here is a quick example of the multiply/divide pair. Suppose 2x = 10 (two times the number is 10). The x is being multiplied by 2, so to undo it we divide both sides by 2:

2x ÷ 2 = 10 ÷ 2

x = 5

And if x ÷ 3 = 4 (the number divided by 3 is 4), the x is being divided by 3, so we multiply both sides by 3:

x ÷ 3 × 3 = 4 × 3

x = 12

Solving two-step equations

Some equations need two moves, not one. Look at 2x − 3 = 7. Here two things are happening to x: it is multiplied by 2, and 3 is taken away.

We peel these off one at a time, like peeling layers. A good order is: first deal with the + or − part, then deal with the × or ÷ part.

Step 1 — get rid of the − 3. Undo it by adding 3 to both sides:

2x − 3 + 3 = 7 + 3

2x = 10

Step 2 — get rid of the × 2. Undo it by dividing both sides by 2:

2x ÷ 2 = 10 ÷ 2

x = 5

So x = 5. Figure 7.3 shows the same two steps drawn as scales, so you can see that the balance never tips.

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Let’s do one more two-step equation as a worked example. Watch the same two-move plan in action.

Sometimes the unknown sits on both sides, like 6y + 7 = 4y + 21. Don’t panic — the golden rule still works. First, gather all the y’s onto one side. Subtract 4y from both sides so the right side loses its y’s:

6y + 7 − 4y = 4y + 21 − 4y

2y + 7 = 21

Now it is an ordinary two-step equation. Subtract 7 from both sides to get 2y = 14, then divide both sides by 2 to get y = 7.

Checking your answer

Here is a wonderful thing about equations: you can always check if your answer is right, all by yourself. No need to ask anyone.

To check, take the value you found and put it back in place of the letter in the original equation. Then work out the LHS. If the LHS comes out equal to the RHS, your answer is correct. If it doesn’t, you made a slip somewhere and can try again.

Let’s check our answer x = 5 for the equation 2x − 3 = 7. We put 5 wherever x was, on the left side. Figure 7.4 walks through it.

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Working it out: LHS = 2 × 5 − 3 = 10 − 3 = 7. The RHS is 7. They are equal, so x = 5 is correct. Always check — it catches mistakes before they cost you marks.

Word problems → equations

Now the real power. Many everyday problems become easy once you turn them into an equation. The plan is always the same:

1. Name the unknown with a letter (say what it stands for).
2. Write the equation that matches the words.
3. Solve it using the golden rule.
4. Check, and answer in words.

Here is a classic. Madhubanti is planning a party. Each plate of snacks costs ₹25. The shop also charges a fixed ₹50 for delivery. She can spend ₹500 in total. How many plates can she buy?

Let’s turn the words into an equation. Figure 7.5 shows how each phrase becomes a part of the equation.

Tap to zoom

So our equation is:

25p + 50 = 500

Now solve it, step by step.

Quick check: 25 × 18 + 50 = 450 + 50 = 500. Correct, it matches the ₹500 she had. The equation did the hard thinking for us.

Common Mistakes

Quick Check

Practice Problems

Easy

Medium

Challenge

Summary

• An equation is a maths sentence with an ”=” sign that says the left side (LHS) and the right side (RHS) are equal — like a balanced weighing scale.
• A letter-number (x, y, n) stands for an unknown number. Solving the equation means finding the value of that letter which makes LHS equal to RHS.
• To turn a situation or word problem into an equation: name the unknown with a letter, then write what the words say as a maths sentence.
• The golden rule: whatever you do to one side, do the exact same thing to the other side. This keeps the scale balanced. You may add, subtract, multiply, or divide both sides by the same number.
• To get the letter alone, undo what is done to it using the opposite operation: + undone by −, − by +, × by ÷, and ÷ by ×.
• For a two-step equation, peel off one layer at a time — usually deal with the + / − part first, then the × / ÷ part.
• A term that crosses the ”=” sign must change to its opposite (a + becomes −, a × becomes ÷), because you are really undoing it on both sides.
• Always check: put your answer back in place of the letter and confirm the LHS equals the RHS.

What’s Next

Take a moment — you have reached the end of Class 7 Maths. Well done!

Look at how far you have come. You started this course finding patterns and working with numbers, and now you can take a real-life puzzle, name the unknown, write a clean equation, and solve it with confidence. That last skill — finding the unknown — is the doorway to algebra, and you will use it in every higher class, in science, and in everyday life whenever you need to work something out backwards.

Every chapter you finished is a tool you now carry. If any idea ever feels rusty, head back to the chapter list and revisit it — the pictures, worked examples, and practice are always there for you. You have built a strong, real understanding of Class 7 Maths. Be proud of that, and carry this same curiosity — always asking why — into whatever you learn next. You’ve got this.