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which mathematic properties are used in these equations

1. 2x+6=10 & 2x+6-6=10-6
2. 3(2x-5)=9 & 6x-15=9
3. 3x=18 & 3z/3 = 18/3
4. 5x-1=13 & 5x-1+1=13+1
5. x/7 = 5 & 7 * x/7 = 5*7

1 answer

In the equations you've provided, several fundamental mathematical properties are used. Here’s a breakdown for each pair of equations:

  1. Equation: \(2x + 6 = 10\) & Rearrangement: \(2x + 6 - 6 = 10 - 6\)

    • Properties Used:
      • Subtraction Property of Equality: If you subtract the same number from both sides of an equation, the equality is maintained.

  2. Equation: \(3(2x - 5) = 9\) & Expansion: \(6x - 15 = 9\)

    • Properties Used:
      • Distributive Property: This property allows you to multiply a single term by two or more terms inside a set of parentheses.

  3. Equation: \(3x = 18\) & Rearrangement: \(\frac{3z}{3} = \frac{18}{3}\)

    • Properties Used:
      • Division Property of Equality: Dividing both sides of an equation by the same non-zero number does not change the equality.

  4. Equation: \(5x - 1 = 13\) & Rearrangement: \(5x - 1 + 1 = 13 + 1\)

    • Properties Used:
      • Addition Property of Equality: Adding the same number to both sides of an equation maintains the equality.

  5. Equation: \(\frac{x}{7} = 5\) & Rearrangement: \(7 \cdot \frac{x}{7} = 5 \cdot 7\)

    • Properties Used:
      • Multiplication Property of Equality: Multiplying both sides of an equation by the same non-zero number preserves the equality.

In summary, the properties commonly employed in these equations are the Addition Property of Equality, Subtraction Property of Equality, Multiplication Property of Equality, Division Property of Equality, and the Distributive Property.