1. Lesson 1.1 Objective: To solve one step equations Essential Question: How does understanding inverse operations help solve one step equations? Are operations that undo each other such as addition and subtraction. Inverse operations: Isolating the Variable: Vocab: means to get the variable by itself.

2. Properties of Equality Addition/Subtraction Property of Equality: Adding or Subtracting the same number from each side of an equation produces an equivalent equation. If a = b, then a + c = b + c If a = b then a – c = b – c Multiplication/ Division Property of Equality: Multiplying or Dividing the same number from each side of an equation produces an equivalent equation. If a = b, then a ∙ c = b ∙ c If a = b, then a ÷ c = b ÷ c c ≠ 0

3. EX: x + 5 = – 13 subtract 5 from both sides – 5 – 5simplify x = – 18 EX: – 6 = x – 9 check add 9 to both sides + 9 +9simplify 3 = x check – 6 = 3 – 9 – 18 + 5 = – 13

4. EX: 7 – x = 42 subtract 7 from both sides – 7 – x = 35 we are solving for x not – x. +x +x 0 = 35 + x subtract 35 from both sides – 35 –35 add x to both sides – 35 = xcheck 7 – (– 35) =42 We could of solved – x = 35 by dividing both sides by –1

5. The temperature in Anchorage Alaska fell from 17degrees at 6:00p.m. to – 6 degrees at 6:00a.m. Find how many degrees the temperature fell. T = temperature change 17degrees + temperature change = new temperature 17 + T = – 6 Subtract 17 from both sides – 17 T = – 23 The temperature fell 23 degrees

6. EX: solve x = 32 – 4 Multiply both sides by – 4 (– 4) x –4 = 32 The -4’s cancel x = – 128 check – 128 – 4 = 32 (– 4) EX: solve 6 = – 2 — 3 x Multiply each side by – 3 — 2 Why 3 – — 2 ? The reciprocal of –2/3 is – 3/2 – 3 2 6 = – 2 3 – 3 2 x – 9 = x Check: - 2/3 –9 = 6

7. Practice: Solve each equation 1) –3x = 27 2) 3) – 3 – 3 X = –9 X= 40 X = 32

8. Discrete math: Vocabulary: Hamiltonian Circuit/Paths: A Hamiltonian path in a graph is a path that passes through every vertex in the graph exactly once. A Hamiltonian path does not necessarily pass through all the edges of the graph, however. A Hamiltonian path which ends in the same place in which it began is called a Hamiltonian circuit.

9. Trace a Hamiltonian path Only a path, not a circuit. The path did not end at the same vertex it started. The path does not need to go over every edge but it can only go over an edge once and must pass through every vertex exactly once. Hamiltonian Circuits are often called the mail man circuit because the mailman goes to every mailbox but does not need to go over every street.

10. A weighted edge has a value assigned to the edge (i.e.- miles, time, gallons of gas, any unit of measure) N 6 5 2 3 5 4 3 4 6 4 6 4 Bank Pharmacy Post Office Dry Cleaners Grocery Store Home Sweet Home Video Store Find the quickest route from home to the pharmacy

11. Review: Euler GraphsPasses over edge exactly once. May pass through a vertex more than once. Hamiltonian Graphs Passes through every vertex exactly once but not necessarily over every edge. Circuits The path ends at the same vertex it started.