Equations · Technique guide
How to Solve Equations in Your Head
By Robert Hansson, Kung Fu Math
An equation is a riddle: some number has been hidden behind a letter, and your job is to find it. On paper we write line after line of algebra, but most everyday equations can be solved faster in your head once you see what the equation is really saying.
The core idea is simple. Whatever was done to x, undo it in reverse order. The techniques below give you different ways to see that undoing quickly, from the most basic linear equation to simple quadratics.
01
Work backwards
Read the equation as a recipe. In 3x + 7 = 31, the recipe says: take x, multiply by 3, add 7, and you get 31. To find x, run the recipe backwards: start at 31, undo the +7, then undo the ×3.
This is the fastest method for equations where x appears only once. You never write anything down; you simply walk back along the chain of operations.
Worked example
3x + 7 = 31
- → Start at 31
- → Undo + 7: 31 − 7 = 24
- → Undo × 3: 24 ÷ 3 = 8
- → Check: 3 · 8 + 7 = 31
Answer: 8
Practice: Try 5x − 4 = 41, 2x + 13 = 47 and (x + 6) ÷ 4 = 9.
02
Keep the balance
When x appears on both sides, picture a balance scale. You may remove the same thing from both pans and the scale stays level. Move all x terms to the side with more x, and all plain numbers to the other side.
Choose the direction that keeps the numbers positive. That one small habit removes most sign errors.
Worked example
7x − 5 = 3x + 19
- → Remove 3x from both sides: 4x − 5 = 19
- → Add 5 to both sides: 4x = 24
- → Divide by 4: x = 6
Answer: 6
Practice: Try 9x + 2 = 4x + 37, 6x − 8 = 2x + 12 and 11x = 5x + 42.
03
Clear the fractions first
Fractions make equations look harder than they are. Multiply every term by the common denominator and the fractions disappear. Then solve the friendly equation that remains.
For a single fraction, cross the denominator over: x ÷ 4 = 7 means x = 28 straight away.
Worked example
x/3 + x/6 = 5
- → The common denominator is 6
- → Multiply every term by 6: 2x + x = 30
- → 3x = 30
- → x = 10
Answer: 10
Practice: Try x/2 + x/4 = 9, x/5 − 2 = 3 and 2x/3 = 14.
04
Use the zero product rule
If a product equals zero, at least one factor must be zero. So (x − 4)(x + 9) = 0 has the solutions x = 4 and x = −9 with no calculation at all. You simply ask: what makes each bracket zero?
Many equations can be brought into this form by factoring out x: x² − 5x = 0 becomes x(x − 5) = 0, so x = 0 or x = 5.
Worked example
x² − 12x = 0
- → Factor out x: x(x − 12) = 0
- → Either x = 0
- → or x − 12 = 0, so x = 12
Answer: 0 and 12
Practice: Try x² + 7x = 0, (2x − 6)(x + 1) = 0 and 3x² = 15x.
05
Spot the factors of a quadratic
For x² + bx + c = 0, look for two numbers that multiply to c and add to b. If you find them, the quadratic factors and the zero product rule finishes the job.
For x² − 5x + 6 = 0 you need two numbers with product 6 and sum −5: that is −2 and −3, so x = 2 or x = 3. With practice you will spot these pairs as quickly as a times table.
Worked example
x² − 9x + 20 = 0
- → Product 20, sum −9: the pair is −4 and −5
- → (x − 4)(x − 5) = 0
- → x = 4 or x = 5
Answer: 4 and 5
Practice: Try x² − 7x + 12 = 0, x² + x − 6 = 0 and x² − 10x + 21 = 0.
Common mistakes
- Undoing operations in the wrong order. Undo the last thing that was done to x first.
- Doing something to only one side of the equation, or only to some of the terms.
- Forgetting the second solution of a quadratic, especially x = 0.
- Skipping the check. Putting your answer back into the original equation takes seconds.
Summary
- Read the equation as a recipe and run it backwards.
- Collect x terms on the side with more x to keep numbers positive.
- Clear fractions by multiplying everything by the common denominator.
- A product equals zero only when one of its factors is zero.
Into the dojo
Reading builds understanding; practice builds speed. Train this in Kung Fu Math — no account needed.