Solving linear equations with a balance model

• Interpret an equals sign as a statement that two expressions have the same value.
• Use inverse operations on both sides of an equation while preserving equality.
• Verify a proposed solution by substituting it into the original equation.

Build the idea

Worked example

1. Step 1

   Subtract five from each side. The left side loses its five unit tiles, while the right side changes by the same amount, so equality is preserved.

   $$3x + 5 - 5 = 20 - 5$$

2. Step 2

   Combine the additive inverse pair on the left and calculate the subtraction on the right. This produces three equal copies of x with total value fifteen.

   $$3x = 15$$

3. Step 3

   Divide both sides by three. The three copies of x become one copy, and fifteen divided into three equal groups gives five in each group.

   $$\frac{3x}{3} = \frac{15}{3}$$

4. Step 4

   After simplifying both quotients, the isolated variable has value five. The result is still a claim until it is checked against the original equation.

   $$x = 5$$

• Moving the five across the equals sign and changing its sign without explaining why.

  The reliable operation is subtracting five from both sides. The apparent sign change is only a shorthand description of that equality-preserving action, not a rule that terms can jump across a symbol.

• Dividing only the three by three while leaving another term on the same side unchanged.

  Division applies to the entire expression on each side. Isolate the product three x before dividing, or place the whole side over the divisor so every term is treated consistently.

Use the same balance reasoning to solve four x minus seven equals thirteen, then verify your value in the original equation.

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

1. Part 1

   What the balance represents

   Start with the equation three x plus five equals twenty. Read the equals sign as a balance point: three identical unknown boxes and five unit tiles have the same total weight as twenty unit tiles. Our goal is not to move symbols arbitrarily. It is to simplify both pans without changing that relationship.

   • Three times x, plus five, equals twenty.
2. Part 2

   Undo the added five

   The five unit tiles are the outer layer around the variable term, so remove five tiles from each pan. On the symbolic line, subtract five from both sides. The left simplifies to three x, and the right simplifies to fifteen. Because both pans changed equally, the new equation has the same solution as the first one.

   • Three times x plus five minus five equals twenty minus five.
   • Three times x equals fifteen.
3. Part 3

   Divide into equal groups

   Three copies of x together weigh fifteen units. Split both sides into three equal groups. Each left group contains one x, while each right group contains five units. This division isolates the variable and gives x equals five without breaking the balance.

   • Three times x divided by three equals fifteen divided by three.
   • x equals five.
4. Part 4

   Verify the result

   Return to the original equation rather than checking only the last line. Replace x with five. Three times five plus five becomes twenty, which matches the original right side. That true statement confirms the value. If the two sides had differed, we would revisit the arithmetic instead of forcing the answer.

   • Three times x, plus five, equals twenty.
   • x equals five.

Scope and limitations

• This lesson covers one-variable linear equations with numerical coefficients, not inequalities, absolute-value equations, or equations with variable denominators.
• The balance picture explains legal transformations but does not by itself address cases with no solution or infinitely many solutions.