Solving Linear Equations
Grade 6 Mathematics — Algebra
📐 1. What is a Linear Equation?
Definition:
A linear equation is an equation where the variable (usually x) has a power of 1. This means we only see x, never x², x³, or higher powers.
Standard Forms:
- ax + b = c — variable on one side
- ax + b = cx + d — variables on both sides
Examples of Linear Equations:
✅ 2x + 5 = 11
✅ 3x − 4 = x + 6
✅ x/2 = 8
NOT Linear Equations:
❌ x² + 3 = 12 (has x²)
❌ x³ = 27 (has x³)
Linear vs Non-Linear Graphs
Linear equations always graph as straight lines!
⚖️ 2. The Main Principle: Balance!
First, without x: what the sign “=” says
The sign “=” says: the left side and the right side are the same number.
Take a true equality 8 + 4 = 7 + 5. The left side is 12. The right side is 12 too.
- Subtract 5 from both sides: 8 + 4 − 5 = 7 + 5 − 5, that is 7 = 7. The equality is still true.
- Add 3 to both sides: 8 + 4 + 3 = 7 + 5 + 3, that is 15 = 15. True again.
Why? Both sides were the same number, 12. If you do the same thing to the same number, you get the same result. So the two sides stay equal.
This is the property of equality. You may add the same number to both sides. You may subtract the same number from both sides. You may multiply or divide both sides by the same number that is not zero.
The scale was balanced. We took 5 from each pan, and it stayed balanced.
Where this does not work
- Subtract different numbers from the sides: 8 + 4 − 5 = 7 + 5 − 2 gives 7 = 10. That is false.
- Add to only one side: 8 + 4 + 3 = 7 + 5 gives 15 = 12. That is false.
- You cannot divide by zero: 12 ÷ 0 has no value. So both sides may be divided only by a number that is not zero.
The same task from primary school
In primary school there was a task: 8 + 4 = □ + 5. Write x instead of the box: 8 + 4 = x + 5.
- Work out the left side: 12 = x + 5.
- Subtract 5 from both sides: 12 − 5 = x + 5 − 5, that is 7 = x.
- Check: 7 + 5 = 12 and 8 + 4 = 12. Both sides are equal.
Where “move a term to the other side and change its sign” comes from
For x + 5 = 12 people say: “move 5 to the right side and change its sign” — and write x = 12 − 5. This is a short record of the same step: we subtracted 5 from both sides.
x + 5 − 5 = 12 − 5 → x = 12 − 5 → x = 7
On the left, +5 and −5 together give 0. On the right, 12 − 5 is left. That is why the term “moves” to the other side with the other sign.
Sometimes x + 5 = 12 is “solved” as x = 12. This happens when “=” is read as “write the answer”. The sign “=” says that both sides are equal. So x + 5 equals 12, and x is 5 less: x = 12 − 5 = 7.
🔑 Golden Rule:
Whatever you do to one side of the equation, you MUST do to the other side!
Our Goal:
Get the variable alone on one side of the equation. We call this isolating the variable.
How We Do It:
- Use inverse operations to undo what's done to the variable
- Keep the equation balanced
- Work step by step until x is alone
Think of an equation as a balanced scale — both sides must stay equal!