Learn›Mathematics KnowledgeMK›Solving equations
Solving equations
Find x. It is the most common question on Mathematics Knowledge, and every version uses the same idea: keep the two sides balanced while you peel away everything around x, in the reverse order it was put on.
An equation is a balance
- 4x + 9 = 37
- 4x + 9 − 9 = 37 − 9
- 4x = 28
- 4x ÷ 4 = 28 ÷ 4
- x = 7
The equals sign says the two sides weigh the same. You can add, subtract, multiply or divide, as long as you do exactly the same thing to both sides: the balance stays level and the equation stays true.
Undo in reverse order
To build 4x + 9, you multiply x by 4 and then add 9. To get x back, undo those in the opposite order: first take away the 9, then divide by 4. Addition and subtraction come off first; multiplication and division last.
How to solve them
-
1
One-step equations: undo the one operation
Each operation has an opposite: subtraction undoes addition, division undoes multiplication, and the other way round.
Next step: Two stepsUndo + with −x + 13 = 31
x = 31 − 13 = 18
Undo × with ÷6x = 54
x = 54 ÷ 6 = 9
Undo ÷ with ×x ÷ 5 = 12
x = 12 × 5 = 60
-
2
Two-step equations: + and − first, then × and ÷
First move the number that is added or subtracted. Then divide (or multiply) to free x.
Next step: Both sides- 5x − 8 = 27
- 5x = 27 + 8
- 5x = 35
- x = 35 ÷ 5
- x = 7
-
3
x on both sides: gather the x terms on one side
Subtract the smaller x term from both sides so that x appears only once. Then it is a two-step equation.
Next step: Parentheses- 7x + 4 = 3x + 24
- 7x − 3x + 4 = 24
- 4x + 4 = 24
- 4x = 24 − 4
- x = 20 ÷ 4
- x = 5
-
4
Parentheses: distribute to every term first
Multiply the number outside by each term inside. Forgetting the second term is the most common slip here.
Next step: Fractions- 3(x − 2) = 2x + 5
- 3x − 6 = 2x + 5
- x − 6 = 5
- x = 5 + 6
- x = 11
-
5
Fractions: multiply every term by the bottom number
Multiply both sides, every single term, by the denominator (or by a number all the denominators go into). The fractions disappear and you are left with a plain equation.
Next step: Check it- x/4 + 3 = 8
- 4 × x/4 + 4 × 3 = 4 × 8
- x + 12 = 32
- x = 20
ShortcutTwo different denominators
x/3 + x/6 = 5: both 3 and 6 go into 6, so multiply every term by 6. That gives 2x + x = 30, so 3x = 30 and x = 10.
-
6
Plug the answer back into the original equation
Put your value of x into the equation you started with, not into one of your steps. Both sides must come out equal. Ten seconds of checking catches almost every sign error.
ExampleCheck x = 5 in 7x + 4 = 3x + 24
- Left side: 7 × 5 + 4 = 35 + 4 = 39.
- Right side: 3 × 5 + 24 = 15 + 24 = 39.
- Both sides are 39, so x = 5 is right.
Answer Checked
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Changing only one side
2x + 6 = 20:
Wrong2x = 20 → x = 10
Right2x = 20 − 6 = 14 → x = 7
Undoing in the wrong order
3x + 12 = 30:
Wrongdivide first: x + 12 = 10
Right3x = 18 → x = 6
Distributing to the first term only
4(x + 3) = 28:
Wrong4x + 3 = 28
Right4x + 12 = 28 → x = 4
Losing the sign when a term moves
9x = 5x − 16:
Wrong4x = 16 → x = 4
Right4x = −16 → x = −4
5 questions at test level
Pick an answer, then read why. No timer, no account.
Solve for x: x − 17 = 29
Show the answer
C. Undo the subtraction by adding 17 to both sides: x = 29 + 17 = 46. Check: 46 − 17 = 29. 12 subtracts instead of adding. x is 46.
Solve for x: 4x + 11 = 47
Show the answer
A. Subtract 11 from both sides: 4x = 36. Divide by 4: x = 9. 36 stops at 4x, and 11.75 divides before subtracting. x is 9.
Solve for x: 8x − 5 = 3x + 30
Show the answer
C. Subtract 3x from both sides: 5x − 5 = 30. Add 5: 5x = 35. Divide: x = 7. 5 subtracts the 5 instead of adding it (5x = 25). x is 7.
Solve for x: 2(3x − 4) = 4x + 10
Show the answer
B. Distribute to both terms: 6x − 8 = 4x + 10. Subtract 4x: 2x − 8 = 10. Add 8: 2x = 18, so x = 9. 7 multiplies only the 3x (6x − 4), and 18 stops at 2x. x is 9.
Solve for x: x/3 + x/4 = 14
Show the answer
A. Multiply every term by 12 (3 and 4 both go into it): 4x + 3x = 168, so 7x = 168 and x = 24. Check: 8 + 6 = 14. 42 and 56 clear only one fraction. x is 24.
10 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 10 cards known.
You finished solving equations.
Practice score: not taken yet Take the practice
Next topic: Exponents & rootsSee this topic move.
In the course, every topic is drawn step by step on a whiteboard while a voice explains it, with timed practice tests and a plan for your test date. The first lessons of every module are free.
