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Understanding Linear Equations: Addition and Multiplication Properties

Learn how to solve linear equations in one variable using the addition and multiplication properties of equality. Equivalent equations, simplification, and solving examples provided.

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Understanding Linear Equations: Addition and Multiplication Properties

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  1. The Addition and Multiplication Properties of Equality § 2.2

  2. Linear Equations Linear equations in one variable can be written in the form ax + b = c, where a, b and c are real numbers, and a 0. Equivalent equations are equations that have the same solution.

  3. 8 + z = – 8 a.) 8 + (– 8) + z = – 8 + – 8 Add –8 to each side. Addition Property of Equality Addition Property of Equality If a, b, and c are real numbers, then a = b and a + c = b + c are equivalent equations. Example: z = – 16 Simplify both sides.

  4. 3p + (– 2p) – 11 = 2p + (– 2p) – 18 Add –2p to both sides. p – 11 + 11 = – 18 + 11 Add 11 to both sides. Solving Equations Example: 4p – 11 – p = 2 + 2p – 20 3p – 11 = 2p – 18 (Simplify both sides.) p – 11 = – 18 Simplify both sides. p = – 7 Simplify both sides.

  5. 6 – 3z + 4z = – 4z + 4zAdd 4z to both sides. 6 + (– 6) + z = 0 +( – 6) Add –6 to both sides. Solving Equations Example: 5(3 + z) – (8z + 9) = – 4z 15 + 5z – 8z – 9 = – 4zUse distributive property. 6 – 3z = – 4zSimplify left side. 6 + z = 0 Simplify both sides. z = – 6 Simplify both sides.

  6. Word Phrases as Algebraic Expressions Example: Write the following sentence as an equation. The product of – 5 and – 29 gives 145. The product of – 5 and – 29 gives 145 In words: Translate: (– 5) · (– 29) = 145

  7. Multiplication Property of Equality (–1)(–y) = 8(–1) Multiply both sides by –1. Multiplication Property of Equality If a, b, and c are real numbers, then a = b and ac = bc are equivalent equations Example: –y = 8 y = –8 Simplify both sides.

  8. Multiply both sides by 7. Simplify both sides. Solving Equations Example:

  9. Multiply both sides by the reciprocal. Simplify both sides. Solving Equations Example:

  10. 3z – 1 + 1 = 26 + 1 Add 1 to both sides. Divide both sides by 3. Using Both Properties Example: 3z – 1 = 26 3z = 27 Simplify both sides. z = 9 Simplify both sides.

  11. 20x + 24 + (–24) = 10 + (–24) Add –24 to both sides. Divide both sides by 20. Simplify both sides. Using Both Properties Example: 12x + 30 + 8x – 6 = 10 20x + 24 = 10 Simplify the left side. 20x = –14 Simplify both sides.

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