Linear Equations
“Solving an equation is 'strand x alone without tipping the scale'.”
The formula
ax + b = 0 → x = −b/aHow to read it: the solution of the linear equation in x is x = −b/a (with a ≠ 0)
- x
- — the unknown you're solving for
- a, b
- — given numbers (a ≠ 0)
- x = −b/a
- — the single value that makes the equation true
The hook
An equation is a riddle: 'what value of x keeps this scale balanced?'
In plain words
An equality that's true for exactly one value of x. Finding that value is what 'solving the equation' means.
The intuition
Picture a balance scale. Add, subtract, multiply, or divide by the same thing on both sides and the balance holds. Use that to strand x alone on one side and the answer shows itself.
How it's built
It has the form ax+b=0. Move b to the other side (transpose) and divide both sides by a to get x=−b/a. Transposing is really a shortcut for 'subtract the same number from both sides'.
Example
2x−6=0 → send 6 to the right: 2x=6 → divide both sides by 2: x=3. Check: 2·3−6=0, true.
Common mistake
When you transpose, the sign flips — a +6 becomes −6 as it crosses over. Carrying the sign unchanged is the most common slip.
Where it's used
Any problem that 'works backward from a condition to one unknown' — change owed, ages, speed, concentration. It's the skeleton of them all.
Where it came from
The word 'algebra' comes from al-Khwarizmi's 9th-century book title 'al-jabr' — the very rule for restoring and transposing terms.
Prerequisites
Quick check
What is the solution of 4x + 12 = 0?
- −3✓
- 3
- −12
- 12
Practice
Solve x + 5 = 12.
Answer: 7
- Transpose 5 to the right: x = 12 − 5
- x = 7
Takeaway: Transposing flips the sign.
Solve 3x − 4 = 2x + 5.
Answer: 9
- Gather x's and numbers: 3x − 2x = 5 + 4
- x = 9
Takeaway: Unknowns to one side, numbers to the other.
Solve 2(x − 1) = x + 4.
Answer: 6
- Expand: 2x − 2 = x + 4
- Transpose: 2x − x = 4 + 2
- x = 6
Takeaway: Expand the brackets first, then transpose.
Three times a number plus 4 is 19. Set up an equation and find the number.
Answer: undefined
- Let the number be x: 3x + 4 = 19
- 3x = 15
- x = 5
Takeaway: Translating the sentence into an equation is half the battle.