Balance Principle Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Balance Principle.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The rule that any operation applied to one side of an equation must also be applied to the other side to preserve equality.

An equation is like a balanced scaleβ€”add weight to both sides equally.

Read the full concept explanation β†’

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The balance principle keeps an equation true by applying any operation equally to both sides of the equals sign.

Common stuck point: The procedure for balance principle is the easy part; the trap is operating on only one side. Asking "Am I applying the identical operation to both sides to preserve equality?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I applying the identical operation to both sides to preserve equality?

Worked Examples

Example 1

easy
Solve x+7=15x + 7 = 15 using the balance principle.

Answer

x=8x = 8

First step

1
An equation is like a balance scale β€” both sides must stay equal.

Full solution

  1. 2
    To isolate xx, subtract 7 from both sides.
  2. 3
    x+7βˆ’7=15βˆ’7x + 7 - 7 = 15 - 7.
  3. 4
    x=8x = 8.
  4. 5
    Check: 8+7=158 + 7 = 15 βœ“
The balance principle: whatever you do to one side, do to the other. Subtract 7 from both sides to keep the equation balanced.

Example 2

medium
Solve 3xβˆ’4=143x - 4 = 14 using the balance principle, showing all steps.

Example 3

medium
Solve 5x+8=335x + 8 = 33, showing each balanced step.

Example 4

medium
Solve 2(x+4)=222(x + 4) = 22.

Example 5

medium
Solve βˆ’3x+11=2-3x + 11 = 2 step by step.

Example 6

hard
Solve 5(xβˆ’2)=3(x+4)5(x - 2) = 3(x + 4).

Example 7

hard
Solve 7βˆ’2(3βˆ’x)=5xβˆ’97 - 2(3 - x) = 5x - 9.

Example 8

hard
Carlos has \$8 more than twice what Maya has. Together they have \$50. How much does each have?

Example 9

hard
Solve ∣xβˆ’4∣=9|x - 4| = 9 for all real xx.

Example 10

challenge
Solve for xx: 3x+2xβˆ’1=5\frac{3x + 2}{x - 1} = 5, xβ‰ 1x \ne 1.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Solve nβˆ’9=6n - 9 = 6 using the balance principle.

Example 2

medium
Solve x4+3=10\frac{x}{4} + 3 = 10.

Example 3

easy
Solve x+5=12x + 5 = 12.

Example 4

easy
Solve xβˆ’4=9x - 4 = 9.

Example 5

easy
Solve 3x=183x = 18.

Example 6

easy
Solve x4=5\frac{x}{4} = 5.

Example 7

easy
To keep x=7x = 7 true after adding 33 to the left, what must you do to the right?

Example 8

easy
Is subtracting 22 from only the left side of x+2=6x + 2 = 6 valid?

Example 9

easy
Solve x+10=10x + 10 = 10.

Example 10

easy
What operation undoes 'multiply by 55'?

Example 11

medium
Solve 2x+3=92x + 3 = 9.

Example 12

medium
Solve 5xβˆ’4=165x - 4 = 16.

Example 13

medium
Solve x3+2=5\frac{x}{3} + 2 = 5.

Example 14

medium
From 2x+3=92x + 3 = 9, a student subtracts 33 to get 2x=92x = 9. What went wrong, and what is correct?

Example 15

medium
Solve 4(xβˆ’1)=124(x - 1) = 12.

Example 16

medium
Solve 7=2xβˆ’17 = 2x - 1.

Example 17

medium
Solve 3x+5=x+133x + 5 = x + 13.

Example 18

challenge
Solve 2xβˆ’13=5\frac{2x - 1}{3} = 5 and state each balancing step.

Example 19

challenge
For what value of cc does 2x+c=2x+72x + c = 2x + 7 hold for ALL xx? Explain using balance.

Example 20

challenge
A student solves x2=6\frac{x}{2} = 6 by writing x=6Γ·2=3x = 6 \div 2 = 3. Diagnose the error and give the correct balance step.

Example 21

medium
Solve 6x+2=4x+106x + 2 = 4x + 10.

Example 22

medium
Solve 2(x+3)=142(x + 3) = 14 by first distributing.

Example 23

easy
Solve x+9=17x + 9 = 17.

Example 24

easy
Solve xβˆ’6=11x - 6 = 11.

Example 25

easy
Solve 7x=567x = 56.

Example 26

easy
Solve x5=9\frac{x}{5} = 9.

Example 27

easy
Solve x+0=8x + 0 = 8.

Example 28

easy
Solve 2x=102x = 10.

Example 29

medium
Solve 4xβˆ’7=214x - 7 = 21.

Example 30

medium
Solve x3βˆ’5=4\frac{x}{3} - 5 = 4.

Example 31

medium
Solve 3x+5=x+193x + 5 = x + 19.

Example 32

medium
Solve 2x+13=5\frac{2x + 1}{3} = 5.

Example 33

medium
A scale balances when one side has 3x3x grams and the other has 2424 grams. Find xx.

Example 34

hard
Solve xβˆ’34+x6=3\frac{x - 3}{4} + \frac{x}{6} = 3.

Example 35

hard
Solve 23x+4=12x+7\frac{2}{3}x + 4 = \frac{1}{2}x + 7.

Example 36

hard
Solve for hh in the formula A=12bhA = \frac{1}{2}bh.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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