Properties of Equality and Solving One-Step Equations

Vocabulary Property In symbols In words Reflexive property of equality a = a A number is equal to itself. Symmetric property If a = b, then b = a. If numbers are equal, they will still be equal if the order is changed. Transitive property If a = b and b = c, then a = c. If numbers are equal to the same number, then they are equal to each other. Substitution property of equality If a = b, then b may be substituted for a in any expression containing a. If two numbers are equal, then substituting one in for another does not change the equality of the equation.

Property In symbols In words Addition property of equality If a = b, then a + c = b + c. Adding the same number to both sides of an equation does not change the equality of the equation. Subtraction property of equality If a = b, then  a – c = b – c. Subtracting the same number from both sides of an equation does not change the equality of the equation. Multiplication If a = b and  c ≠ 0, then a • c = b • c. Multiplying both sides of the equation by the same number, other than 0, does not change the equality of the equation. Division property c ≠ 0, then  a ÷ c = b ÷ c. Dividing both sides of the equation by the same number, other than 0, does not change the equality of the equation.

Property General rule Specific example Commutative property a + b = b + a a • b = b • a 3 + 8 = 8 + 3 3 • 8 = 8 • 3 Associative property (a + b) + c = a + (b + c) (a • b) • c = a • (b • c) (3 + 8) + 2 = 3 + (8 + 2) (3 • 8) • 2 = 3 • (8 • 2) Distributive property a • (b + c) = a • b + a • c 3 • (8 + 2) = 3 • 8 + 3 • 2

Like Terms Like terms have the same variable or combination of variables, and those variables have the same exponents. Ex. 3 𝑥 2 𝑦 𝑎𝑛𝑑 18 𝑥 2 𝑦 𝑥 𝑎𝑛𝑑 15𝑥

Combining Like Terms 2x To combine like terms add the coefficients. Example. 4𝑥+3𝑥−6𝑥 2x variable

Example Given: 2x - 4y - 3x + 7y Combine like terms.

Example Given: 3 𝑥 2 +9 𝑦 4 −10 𝑥 2 − 𝑦 3 Combine like terms

Solving Equations Solving an equation for a variable means finding the values of the variable that make the equation true. To do this we must isolate the variable to one side of the equation by using inverse operations.

Inverse Operations Inverse operations undo operations; you can think of them as opposites. -Addition and subtraction are inverse operations. -Multiplication and division are inverse operations.

An equation is like a balance scale because it shows that two quantities are equal. What you do to one side of the equation must also be done to the other side to keep it balanced.

Example Given: k + 5 = 17 solve for k.

Example Given: 2x = 18 solve for x.

Example Given: 9 + x = 20 solve for x.