What this chapter is about
This chapter teaches you how to find a number that you do not know yet. We call this number the unknown. In mathematics, we use letters like x, y or n to stand for the unknown. The job is to figure out what value the letter must have so that a given statement becomes true.
You already know how to do operations with numbers. Now you learn to work backwards. If someone tells you that a number plus 5 equals 12, you can think: what number added to 5 gives 12? The answer is 7. This chapter shows you a neat, step-by-step way to find such answers using simple equations.
After studying this chapter, you should be able to write an equation from a word problem, solve the equation to find the unknown, and check that your answer is correct by putting it back into the original statement.
Key ideas
- An equation is a statement that two things are equal. For example, x + 4 = 10 says that some number plus 4 equals 10.
- The unknown is the letter in the equation. Our goal is to find what number it stands for.
- Whatever you do to one side of an equation, you must do the same to the other side. This keeps the two sides equal.
- To isolate the unknown, undo the operation that is done to it. If something is added, subtract it from both sides. If something is multiplied, divide both sides by that number.
- A solution is the value of the unknown that makes the equation true.
- You can always check your answer by substituting it back into the original equation.
Formulas and facts to remember
- If x + a = b, then x = b − a. (Subtract the added number from both sides.)
- If x − a = b, then x = b + a. (Add back what was subtracted.)
- If a × x = b, then x = b ÷ a. (Divide both sides by the multiplier.)
- If x ÷ a = b, then x = b × a. (Multiply both sides by the divisor.)
- Checking: Put the answer back into the equation. Left side should equal right side.
Worked examples
Example 1 Ravi thinks of a number. He adds 9 to it and gets 26. What is his number?
Let the unknown number be n. Equation: n + 9 = 26 Subtract 9 from both sides: n = 26 − 9 = 17 Check: 17 + 9 = 26. Correct. So Ravi's number is 17.
Example 2 A fruit seller has some oranges. She sells 15 and 32 remain. How many did she have at first?
Let the unknown be x (oranges she had). Equation: x − 15 = 32 Add 15 to both sides: x = 32 + 15 = 47 Check: 47 − 15 = 32. Correct. She had 47 oranges.
Example 3 Five identical notebooks cost ₹120 in total. What is the price of one notebook?
Let the price of one notebook be p rupees. Equation: 5 × p = 120 Divide both sides by 5: p = 120 ÷ 5 = 24 Check: 5 × 24 = 120. Correct. One notebook costs ₹24.
Common mistakes
Doing an operation on one side only → Always do the same operation on both sides to keep the equation balanced.
Using the wrong opposite operation, such as adding when you should subtract → Remember: addition undoes subtraction, multiplication undoes division.
Forgetting to check the answer → Substitute your answer back into the original equation to be sure it works.
Writing the equation the wrong way round, like 15 − x = 32 instead of x − 15 = 32 → Read the problem carefully to see what is happening to the unknown.
Mixing up the unknown with the answer too soon → Keep the letter until you finish solving; only then state the value.
Quick revision
- An equation says two expressions are equal; solve it to find the unknown.
- Undo addition by subtracting, and undo subtraction by adding.
- Undo multiplication by dividing, and undo division by multiplying.
- Always perform the same step on both sides.
- Check your answer by substituting it back.