✍ Class 8 Mathematics — Linear Equations in One Variable
Interactive Visual Revision Notes
Aligned with the Class 8 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.
🌎 Real-World Connection: Imagine a balance scale. If you put 4 apples and a 7-gram weight on one side, and 2 apples and a 15-gram weight on the other — to balance it, how heavy is each apple? That is a linear equation: 4x + 7 = 2x + 15. Linear equations help us solve age puzzles, money problems, distance-speed-time questions, and much more. They are the engine behind spreadsheets, coding, and financial planning.
✍ Topic 1: What Is an Equation?
An equation has two sides.
Both sides are joined by an equal sign.
The left side equals the right side.
Example: 3 + 4 = 7 is an equation.
Example: x + 2 = 5 is also an equation.
If we remove the equal sign, it is not an equation.
✍ Topic 2: Variable and Constant
A variable is an unknown value.
We use letters like x, y, or z.
A constant is a fixed number.
Example: In x + 5 = 9, x is the variable.
The numbers 5 and 9 are constants.
We need to find the value of x.
✍ Topic 3: Linear Equation
In a linear equation, the variable has power 1.
That means we see x, not x² or x³.
Example: 2x + 3 = 11 is linear.
Example: x² + 1 = 5 is not linear.
Linear means the graph is a straight line.
We solve for one variable only.
Highest power of the variable = 1
✍ Topic 4: Solving by Balancing Method
Think of an equation as a balance scale.
Both sides must stay equal.
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.
This keeps the balance perfect.
✍ Topic 5: Variables on Both Sides
Sometimes x appears on both sides.
Example: 3x + 5 = 2x + 12.
Move all x terms to one side.
Move all numbers to the other side.
Subtract 2x from both sides: x + 5 = 12.
Subtract 5 from both sides: x = 7.
✍ Topic 6: Transposing Terms
Transposing = moving a term to the other side.
When you move a term, change its sign.
Plus becomes minus. Minus becomes plus.
Example: x + 3 = 10.
Move +3 to the right: x = 10 − 3 = 7.
This is a shortcut for the balancing method.
Move to other side = Change the sign
✍ Topic 7: Equations with Fractions
Some equations have fractions.
Example: x/3 + 2 = 5.
First, subtract 2 from both sides.
We get x/3 = 3.
Then multiply both sides by 3.
We get x = 9.
Tip: Remove fractions early to make it easier.
✍ Topic 8: Cross Multiplication Method
Use this when both sides are fractions.
If a/b = c/d, then a × d = b × c.
This removes all fractions at once.
Example: x/4 = 3/2.
Cross multiply: x × 2 = 4 × 3.
So 2x = 12, which gives x = 6.
a/b = c/d → a × d = b × c
✍ Topic 9: Word Problems
Read the problem carefully.
Pick a variable for the unknown.
Write the problem as an equation.
Age problems: Use x for age now.
Number problems: Use x for the number.
Perimeter problems: Add all sides.
Solve the equation. Then check your answer.
✍ Topic 10: Checking Your Answer
After solving, put the value back into the equation.
This is called substitution.
Check: Does the left side equal the right side?
If yes, your answer is correct.
If no, solve again carefully.
Always check your work to be sure.
🎨 Section 2 — Easy Illustrations
Picture 1: Balance Scale Visual
An equation works like a balance. Both sides weigh the same.
Picture 2: Solving Steps Flowchart
Follow these four steps every time you solve an equation.
Picture 3: Variables on Both Sides
Move all x terms to the left. Move all numbers to the right. Solve!
Picture 4: Cross Multiplication
Cross multiply: top-left times bottom-right = bottom-left times top-right.
Picture 5: Word Problem to Equation
Read the words. Pick a variable. Write an equation. Solve. Check!