Simple Equations
Have you ever played a guessing game where someone says, "I am thinking of a number. If you add 5 to it, you get 12. What is the number?" You would quickly figure out the answer is 7. Without realising it, you just solved an equation! An equation is a mathematical statement that says two things are equal, and solving an equation means finding the value of the unknown number.
Simple equations are one of the most exciting topics in Class 7 Maths because they are the beginning of algebra. Algebra is like a superpower in maths that lets you solve problems by using letters (like x, y, or n) to represent unknown numbers. Once you write a problem as an equation, you can use rules to find the unknown value step by step.
In this chapter, we will learn what an equation is, what variables and constants are, how to form equations from word problems, and how to solve equations using different methods. We will use real-life examples like age puzzles, pocket money problems, and sharing sweets to make the concepts easy to understand.
The key idea is simple: an equation is like a balanced weighing scale. Whatever you do to one side, you must do to the other side to keep it balanced. If the left side equals the right side, and you add 3 to the left, you must add 3 to the right too. This balance principle is the foundation of everything we will learn about equations.
What is Simple Equations?
An equation is a mathematical statement that shows that two expressions are equal. It always has an equals sign (=) separating the left-hand side (LHS) from the right-hand side (RHS).
For example: 2x + 3 = 11 is an equation. The left side is 2x + 3 and the right side is 11.
Key Terms:
| Term | Meaning | Example |
|---|---|---|
| Variable | A letter that represents an unknown number | In 2x + 3 = 11, x is the variable |
| Constant | A fixed number that does not change | In 2x + 3 = 11, the numbers 2, 3, and 11 are constants |
| Coefficient | The number multiplied by the variable | In 2x + 3 = 11, 2 is the coefficient of x |
| LHS | Left-Hand Side of the equation | 2x + 3 |
| RHS | Right-Hand Side of the equation | 11 |
| Solution | The value of the variable that makes LHS = RHS | x = 4 (because 2(4) + 3 = 11) |
What is NOT an equation:
- 2x + 3 > 11 (this is an inequality, not an equation, because it uses > instead of =)
- 2x + 3 (this is just an expression, not an equation, because there is no = sign)
A simple equation (also called a linear equation in one variable) contains only one variable, and the variable has power 1 (not squared, cubed, etc.). Examples: x + 5 = 12, 3y - 7 = 20, 2n/3 = 8.
Simple Equations Formula
Solving Equations - The Balance Method:
Think of an equation as a balanced weighing scale. To keep the balance:
Whatever you do to one side, you must do the same to the other side.
- You can add the same number to both sides.
- You can subtract the same number from both sides.
- You can multiply both sides by the same number.
- You can divide both sides by the same number (not zero).
Solving Equations - The Transposition Method:
Transposition means moving a term from one side of the equation to the other by changing its sign:
+ becomes - when moved across =
- becomes + when moved across =
x becomes / when moved across =
/ becomes x when moved across =
Example: x + 5 = 12 → x = 12 - 5 → x = 7 (the +5 moved to the other side as -5).
Verification:
Always check your answer by substituting it back into the original equation:
Substitute the value into LHS. If LHS = RHS, the solution is correct.
Types and Properties
Simple equations can be categorised based on the operations involved:
Type 1: x + a = b (Addition Equation)
To solve, subtract a from both sides (or transpose a to the other side as -a). Example: x + 7 = 15 → x = 15 - 7 = 8. Think of it as: "What number plus 7 gives 15?"
Type 2: x - a = b (Subtraction Equation)
To solve, add a to both sides (or transpose -a to the other side as +a). Example: x - 4 = 9 → x = 9 + 4 = 13. Think of it as: "What number minus 4 gives 9?"
Type 3: ax = b (Multiplication Equation)
To solve, divide both sides by a. Example: 3x = 21 → x = 21 / 3 = 7. Think of it as: "3 times what number gives 21?"
Type 4: x/a = b (Division Equation)
To solve, multiply both sides by a. Example: x/5 = 4 → x = 4 x 5 = 20. Think of it as: "What number divided by 5 gives 4?"
Type 5: ax + b = c (Two-Step Equation)
First isolate the variable term (subtract b from both sides), then divide by the coefficient. Example: 2x + 3 = 11 → 2x = 11 - 3 = 8 → x = 8 / 2 = 4.
Type 6: ax - b = c (Two-Step with Subtraction)
First add b to both sides, then divide by a. Example: 5x - 10 = 25 → 5x = 25 + 10 = 35 → x = 35 / 5 = 7.
Type 7: Equations with Variable on Both Sides
Move all variable terms to one side and constants to the other. Example: 3x + 5 = x + 13 → 3x - x = 13 - 5 → 2x = 8 → x = 4.
Forming Equations from Word Problems:
The trickiest part is often converting a word problem into an equation. Key steps: (1) Identify the unknown and call it x. (2) Translate the words into mathematical operations. (3) Write the equation. (4) Solve and verify.
Common Word-to-Maths Translations:
| Words | Mathematical Operation | Example |
|---|---|---|
| "more than", "added to", "increased by", "sum of" | Addition (+) | "5 more than x" → x + 5 |
| "less than", "decreased by", "subtracted from" | Subtraction (-) | "7 less than x" → x - 7 |
| "times", "product of", "multiplied by", "twice", "thrice" | Multiplication (x) | "twice x" → 2x |
| "divided by", "half of", "one-third of" | Division (/) | "half of x" → x/2 |
| "is", "gives", "equals", "results in" | Equals (=) | "the result is 15" → = 15 |
Watch out for tricky phrases: "7 less than x" means x - 7 (not 7 - x). "5 subtracted from x" means x - 5. Always think about what the phrase is saying carefully before writing the equation.
Solved Examples
Example 1: Solving x + a = b
Problem: Solve: x + 9 = 17
Solution:
Step 1: To isolate x, subtract 9 from both sides.
Step 2: x + 9 - 9 = 17 - 9
Step 3: x = 8
Verification: LHS = 8 + 9 = 17 = RHS ✓
Answer: x = 8
Example 2: Solving x - a = b
Problem: Solve: y - 6 = 14
Solution:
Step 1: Add 6 to both sides.
Step 2: y - 6 + 6 = 14 + 6
Step 3: y = 20
Verification: LHS = 20 - 6 = 14 = RHS ✓
Answer: y = 20
Example 3: Solving ax = b
Problem: Solve: 4x = 28
Solution:
Step 1: Divide both sides by 4.
Step 2: 4x / 4 = 28 / 4
Step 3: x = 7
Verification: LHS = 4 x 7 = 28 = RHS ✓
Answer: x = 7
Example 4: Solving x/a = b
Problem: Solve: n/3 = 9
Solution:
Step 1: Multiply both sides by 3.
Step 2: (n/3) x 3 = 9 x 3
Step 3: n = 27
Verification: LHS = 27/3 = 9 = RHS ✓
Answer: n = 27
Example 5: Solving a Two-Step Equation
Problem: Solve: 3x + 5 = 20
Solution:
Step 1: Subtract 5 from both sides: 3x + 5 - 5 = 20 - 5 → 3x = 15
Step 2: Divide both sides by 3: 3x / 3 = 15 / 3 → x = 5
Verification: LHS = 3(5) + 5 = 15 + 5 = 20 = RHS ✓
Answer: x = 5
Example 6: Using Transposition
Problem: Solve: 7x - 8 = 34
Solution:
Step 1: Transpose -8 to RHS (it becomes +8): 7x = 34 + 8 = 42
Step 2: Transpose x7 (divide by 7): x = 42 / 7 = 6
Verification: LHS = 7(6) - 8 = 42 - 8 = 34 = RHS ✓
Answer: x = 6
Example 7: Variable on Both Sides
Problem: Solve: 5x + 3 = 2x + 18
Solution:
Step 1: Move variable terms to LHS: 5x - 2x + 3 = 18 → 3x + 3 = 18
Step 2: Move constant to RHS: 3x = 18 - 3 = 15
Step 3: Divide by 3: x = 15 / 3 = 5
Verification: LHS = 5(5) + 3 = 28. RHS = 2(5) + 18 = 28. LHS = RHS ✓
Answer: x = 5
Example 8: Word Problem: Age Puzzle
Problem: Anu is 5 years older than her brother Binu. If Anu is 17 years old, how old is Binu? Write an equation and solve.
Solution:
Step 1: Let Binu's age = x years.
Step 2: Anu is 5 years older, so Anu's age = x + 5.
Step 3: We know Anu's age = 17. So: x + 5 = 17
Step 4: Solve: x = 17 - 5 = 12
Verification: Binu = 12, Anu = 12 + 5 = 17 ✓
Answer: Binu is 12 years old.
Example 9: Word Problem: Sharing Sweets
Problem: A box has some sweets. When 8 sweets are taken out, 22 sweets remain. How many sweets were in the box originally?
Solution:
Step 1: Let the original number of sweets = x.
Step 2: After taking out 8: x - 8 = 22
Step 3: Solve: x = 22 + 8 = 30
Verification: 30 - 8 = 22 ✓
Answer: The box originally had 30 sweets.
Example 10: Word Problem: Pocket Money
Problem: Rahul's mother gives him some pocket money. He spends Rs. 35 on a notebook and Rs. 20 on a pen, and has Rs. 45 left. How much pocket money did he receive?
Solution:
Step 1: Let pocket money = x rupees.
Step 2: Amount spent = 35 + 20 = Rs. 55. Remaining = Rs. 45.
Step 3: Equation: x - 55 = 45
Step 4: Solve: x = 45 + 55 = 100
Verification: 100 - 55 = 45 ✓
Answer: Rahul received Rs. 100 as pocket money.
Real-World Applications
Simple equations are used everywhere in real life, even if we do not always write them down:
Shopping: "I have Rs. 200. After buying a book, I have Rs. 65 left. How much did the book cost?" This is the equation x + 65 = 200, giving x = 135.
Age Problems: "A father is 30 years older than his son. The father is 45 years old. How old is the son?" Equation: x + 30 = 45, giving x = 15.
Distance and Travel: "A train covers a distance at 60 km/hr in some hours. If the total distance is 240 km, how many hours did it travel?" Equation: 60x = 240, giving x = 4 hours.
Business: "A shopkeeper bought items for Rs. 500 and sold them making a profit of Rs. 150. What was the selling price?" Equation: x - 500 = 150, giving x = 650.
Science: Many scientific formulas are equations. For example, speed = distance/time. If you know the speed and time, you can find the distance using the equation d = s x t.
Puzzles and Games: Number puzzles like "I am thinking of a number; if you double it and add 3, you get 19" are solved using equations: 2x + 3 = 19, giving x = 8.
Cooking: "A recipe needs 3 cups of flour for 12 cookies. How many cups for 20 cookies?" This sets up a proportion equation: 3/12 = x/20.
Key Points to Remember
- An equation is a statement with an equals sign (=) showing that two expressions are equal.
- A variable (like x, y, n) represents an unknown number. A constant is a fixed number.
- A simple equation has one variable with power 1 (linear).
- The solution of an equation is the value of the variable that makes LHS = RHS.
- Balance Method: Whatever operation you perform on one side, perform the same on the other side.
- Transposition: When a term moves across the equals sign, its sign changes (+↔-, x↔/).
- Always verify your answer by substituting it back into the original equation.
- To form equations from word problems: identify the unknown (call it x), translate the problem into mathematical operations, write the equation, solve, and verify.
- Simple equations are the foundation of algebra and are used in higher-level maths extensively.
Practice Problems
- Solve: x + 12 = 25
- Solve: y - 8 = 15
- Solve: 6x = 54
- Solve: m/4 = 7
- Solve: 2x + 7 = 19
- Solve: 4y - 9 = 23
- Solve: 3x + 4 = x + 14
- A number is multiplied by 5 and then 3 is added to the product. The result is 28. Find the number.
- The sum of three consecutive numbers is 72. Find the numbers. (Hint: let the middle number be x.)
- Riya is twice as old as her sister. The sum of their ages is 24 years. Find their ages.
Frequently Asked Questions
Q1. What is a simple equation?
A simple equation (also called a linear equation in one variable) is a mathematical statement with an equals sign, containing one unknown (variable) with power 1. Examples: x + 5 = 12, 3y = 21, 2n - 4 = 10. The goal is to find the value of the variable that makes the equation true.
Q2. What is the difference between an equation and an expression?
An expression is a combination of numbers, variables, and operations (like 2x + 3) but does NOT have an equals sign. An equation has an equals sign showing that two expressions are equal (like 2x + 3 = 11). You can solve an equation to find the variable's value, but you cannot 'solve' an expression.
Q3. What is the balance method for solving equations?
The balance method treats the equation like a weighing scale. To keep the balance, whatever you do to one side, you must do to the other. For example, to solve x + 5 = 12, subtract 5 from both sides: x + 5 - 5 = 12 - 5, which gives x = 7.
Q4. What is transposition?
Transposition is a shortcut for the balance method. When you move a term from one side of the equation to the other, you change its sign. A + becomes -, a - becomes +, multiplication becomes division, and division becomes multiplication. For example, x + 7 = 15 becomes x = 15 - 7 = 8 (the +7 moved to the right as -7).
Q5. How do you check if your answer is correct?
Substitute your answer back into the original equation. If the left-hand side (LHS) equals the right-hand side (RHS), your answer is correct. For example, if x = 4 is the solution to 2x + 3 = 11, check: LHS = 2(4) + 3 = 8 + 3 = 11 = RHS. Correct!
Q6. How do you form an equation from a word problem?
Step 1: Identify the unknown quantity and call it x (or any letter). Step 2: Translate the words into mathematical operations. Step 3: Write the equation using the equals sign. Step 4: Solve the equation. Step 5: Verify the answer. For example, 'A number plus 8 equals 20' becomes x + 8 = 20, so x = 12.
Q7. What are variables and constants?
A variable is a letter (like x, y, n) that represents an unknown number whose value we need to find. A constant is a fixed number that does not change (like 3, 7, -5). In the equation 2x + 5 = 13, x is the variable and 2, 5, and 13 are constants.
Q8. Can an equation have the variable on both sides?
Yes! For example, 3x + 5 = x + 13. To solve, move all variable terms to one side and all constants to the other side: 3x - x = 13 - 5, which gives 2x = 8, so x = 4. Always collect like terms before solving.
