Solving Simple Equations
An equation is like a balance scale — both sides must be equal. If someone tells you "a number plus 5 equals 12", you can write it as x + 5 = 12 and find that the number is 7. This process of finding the unknown value is called solving the equation.
In Class 7 NCERT Maths, you will learn two main methods to solve simple equations: the balance method and the transposition method. Both methods help you find the value of the unknown variable.
A simple equation has only one variable (usually x) and its highest power is 1 (no x², x³, etc.).
What is Solving Simple Equations - Grade 7 Maths (Simple Equations)?
Definition: A simple equation (or linear equation in one variable) is a statement of equality that contains an unknown variable whose value makes the equation true.
Key terms:
- Variable: The unknown quantity, usually written as x, y, or z.
- LHS: Left-Hand Side of the equation.
- RHS: Right-Hand Side of the equation.
- Solution: The value of the variable that makes LHS = RHS.
Solving Simple Equations Formula
Balance Method (Golden Rule):
Whatever you do to one side of the equation, do the same to the other side.
Steps:
- Look at the operation applied to the variable.
- Perform the inverse (opposite) operation on both sides to isolate the variable.
- Addition and subtraction are inverse operations of each other.
- Multiplication and division are inverse operations of each other.
Types and Properties
Types of simple equations you will see:
- Type 1: x + a = b — Solve by subtracting a from both sides.
- Type 2: x − a = b — Solve by adding a to both sides.
- Type 3: ax = b — Solve by dividing both sides by a.
- Type 4: x/a = b — Solve by multiplying both sides by a.
- Type 5: ax + b = c — First subtract b, then divide by a.
- Type 6: Variable on both sides — Bring variable terms to one side and constants to the other.
Solved Examples
Example 1: Solving x + 5 = 12
Problem: Solve: x + 5 = 12
Solution:
Step 1: Subtract 5 from both sides: x + 5 − 5 = 12 − 5
Step 2: Simplify: x = 7
Step 3: Check: 7 + 5 = 12. Correct!
Answer: x = 7
Example 2: Solving x − 8 = 15
Problem: Solve: x − 8 = 15
Solution:
Step 1: Add 8 to both sides: x − 8 + 8 = 15 + 8
Step 2: Simplify: x = 23
Step 3: Check: 23 − 8 = 15. Correct!
Answer: x = 23
Example 3: Solving 3x = 21
Problem: Solve: 3x = 21
Solution:
Step 1: Divide both sides by 3: 3x / 3 = 21 / 3
Step 2: Simplify: x = 7
Step 3: Check: 3 × 7 = 21. Correct!
Answer: x = 7
Example 4: Solving x/4 = 6
Problem: Solve: x/4 = 6
Solution:
Step 1: Multiply both sides by 4: (x/4) × 4 = 6 × 4
Step 2: Simplify: x = 24
Step 3: Check: 24/4 = 6. Correct!
Answer: x = 24
Example 5: Solving 2x + 3 = 15
Problem: Solve: 2x + 3 = 15
Solution:
Step 1: Subtract 3 from both sides: 2x + 3 − 3 = 15 − 3
Step 2: Simplify: 2x = 12
Step 3: Divide both sides by 2: x = 12 / 2 = 6
Step 4: Check: 2(6) + 3 = 12 + 3 = 15. Correct!
Answer: x = 6
Example 6: Solving 5x − 7 = 18
Problem: Solve: 5x − 7 = 18
Solution:
Step 1: Add 7 to both sides: 5x = 18 + 7 = 25
Step 2: Divide both sides by 5: x = 25 / 5 = 5
Step 3: Check: 5(5) − 7 = 25 − 7 = 18. Correct!
Answer: x = 5
Example 7: Equation with Negative Numbers
Problem: Solve: x + 9 = 4
Solution:
Step 1: Subtract 9 from both sides: x = 4 − 9 = −5
Step 2: Check: (−5) + 9 = 4. Correct!
Answer: x = −5
Example 8: Variable on Both Sides
Problem: Solve: 3x + 2 = x + 10
Solution:
Step 1: Subtract x from both sides: 3x − x + 2 = 10
Step 2: Simplify: 2x + 2 = 10
Step 3: Subtract 2 from both sides: 2x = 8
Step 4: Divide by 2: x = 4
Step 5: Check: LHS = 3(4) + 2 = 14. RHS = 4 + 10 = 14. LHS = RHS. Correct!
Answer: x = 4
Example 9: Trial and Error Method
Problem: Solve by trial and error: x + 3 = 10
Solution:
Try x = 5: 5 + 3 = 8 (not 10, too small)
Try x = 7: 7 + 3 = 10 (correct!)
Answer: x = 7
Note: Trial and error works for simple equations but becomes difficult for complex ones. Use the balance method instead.
Example 10: Two-Step Equation with Fraction
Problem: Solve: x/3 + 2 = 7
Solution:
Step 1: Subtract 2 from both sides: x/3 = 7 − 2 = 5
Step 2: Multiply both sides by 3: x = 5 × 3 = 15
Step 3: Check: 15/3 + 2 = 5 + 2 = 7. Correct!
Answer: x = 15
Real-World Applications
Real-life uses of solving equations:
- Shopping: If 3 notebooks cost Rs. 120, the equation 3x = 120 gives the price of one notebook.
- Age problems: "Anu is 5 years older than Riya. Anu is 18" gives the equation x + 5 = 18, so Riya is 13.
- Distance-speed problems: If speed × time = distance, and you know two values, you can find the third.
- Sharing equally: If Rs. 500 is shared among x people and each gets Rs. 100, then 500/x = 100.
- Science: Equations are used in physics formulas like F = ma, V = IR, etc.
Key Points to Remember
- An equation is a statement that two expressions are equal.
- The solution of an equation is the value of the variable that makes it true.
- Balance method: Perform the same operation on both sides to keep the equation balanced.
- Use the inverse operation: addition undoes subtraction, multiplication undoes division.
- Always check your answer by substituting it back into the original equation.
- LHS must equal RHS after substitution for the answer to be correct.
- Simple equations have only one variable with power 1.
- Equations can be solved by trial and error (for simple cases) or by the balance method (for all cases).
Practice Problems
- Solve: x + 11 = 20
- Solve: x − 6 = −3
- Solve: 4x = 32
- Solve: x/5 = 9
- Solve: 3x + 7 = 22
- Solve: 2x − 5 = x + 3
- Solve: x/2 + 4 = 10
- If the sum of a number and 15 is 42, find the number by forming and solving an equation.
Frequently Asked Questions
Q1. What is a simple equation?
A simple equation (or linear equation) is a mathematical statement with an equals sign where the unknown variable has power 1. For example, x + 5 = 12 and 3x = 21 are simple equations.
Q2. What is the balance method?
The balance method means performing the same operation on both sides of the equation. If you add 5 to the left side, you must add 5 to the right side too. This keeps the equation balanced, like a weighing scale.
Q3. Why should we check the solution?
Checking ensures your answer is correct. Substitute your answer back into the original equation. If the left side equals the right side, your solution is correct.
Q4. What is the inverse of multiplication?
Division is the inverse of multiplication. If 3x = 21, you divide both sides by 3 to get x = 7. Similarly, addition is the inverse of subtraction.
Q5. Can the solution of an equation be negative?
Yes. For example, x + 10 = 3 gives x = −7. The solution depends on the equation and can be positive, negative, or zero.
Q6. Can the solution be a fraction or decimal?
Yes. For example, 2x = 5 gives x = 5/2 = 2.5. Not all equations have whole number solutions.
