Chapter 1-2 Equations Study Guide

1. Chapter 1-2 Equations Study Guide

2. Algebraic Identities and Properties • What’s a like term and unlike term? • What are the algebraic properties? • Why do these properties work? • What is a closure?

3. Like Terms and Unlike Terms Important: Make sure you combine all the like terms when simplifying.

4. Commutative, Associative, Distributive, Transitive Property • Commutative - switching orders of multiplication or addition. • Associative - grouping of multiplication or addition • Distributive -multiplicationwith parenthesis • Transitive - logic

5. Do these properties work? • Commutative: • Yes! • Associative: • Yes! • Distributive: • 2(2+3)=2(2)+2(3) 10=4+6 Yes! • Transitive: • 2=2, 2=2, 2=2 Yes!

6. Closure • Definition: a set of numbers is closed if the result of an operation using two numbers in the set is also the set. • Basic meaning: If the operation you’re using on any kind of number (whole numbers, integers …) will also be in that kind, then it is closed under the operation.

7. Examples of Closure • The set of integers is closed under addition. • True! Ex. 2+3=5 • The set of integers is closed under division. • False! Ex. 7/2=3.5 (Integers don’t have decimals!)

8. Solving Equations • How to write an algebraic equation into words? • How to solve equations? • Understanding the Problem Solving Process.

9. Writing Algebraic Equations *Less than reverses the order of subtraction!

10. Solving Equations • Simplify: • Use distributive property, combine like terms • Isolate: • Inverse operations and property of equality • Check (Optional) • Plug in the answer to see if the equation is true.

11. 4-Step Problem Solving Process • Define the variable (x=????) • Write an equation for the situation • Solve the equation • Sentence that answers the question (include units!)

12. Practice Questions • The sum of three consecutive multiples of six is 72. Find the integers. • In triangle ABC, angle A is twice as large as angle B and angle C measures 20° less than double the measure of angle B. Find the measure of each interior angle.

13. Answers • x+x+6+x+12=72 ; x=18 ; 18,24,30 • A=2B ; C=2B-20 ; B=c/2 +10 • A=2B ; A=2(c/2 + 10) ; A=C+20 • (C+20)+(C/2 + 10)+ C=180 • C=60 ; B=40 ; A=80

14. Identities and Contradictions • What are identities and contradictions? • Identifying identities and contradictions.

15. Identity • Definition: When solving an equation, if you get a statement that is always true, it’s called an identity and all real numbers are true. • Basic Meaning: When you solve to the end, and you get something like x=x or 3=3 etc. that means every number is an answer.

16. Contradiction • Definition: When solving an equation, if you get a statement that is never true, the equations are contradicting, then there are no solutions. • Basic Meaning: When you solve to the end and you get something like 2x=3x or 1=2 etc. then there are no answers since 1 can never be 2.

17. Practice • 2(x - 8) + 7=5(x+2) – 3x +9 • 5-3(2x-1) – 1= x – 7(x-1)

18. Answers • -9=19 Contradiction • 7=7 Identity

19. Literal Equations (Formulas) • What are literal equations? • How to solve a literal equation for the variable?

20. Literal Equations • Definition: an equation with two or more variables. • Basic Meaning: when ever you see something like or this is a literal equation.

21. Ratios, Proportions and Rates • What are ratios, and rates? • How to solve a proportion? • What are similar figures?

22. Ratios, Rates • Ratio: comparison of two values with same units. • Ex. Boy: girls = 7:8 • Rate: ratio with different units. • Ex. Miles/h, km/h, $/h …

23. Solving Proportions • Proportion: equation stating two ratios are equal. • Use cross multiply (means-extremes property) to solve a proportion. • In a proportion, cross products are equal

24. Similar Figures • Definition: Figures who have same shape but not necessarily have the same size. • Corresponding angles have same measure • Corresponding sides are in same ratio

25. Conversions • How to convert rates to other rates? • Application of conversions.

26. Converting Rates • Convert by writing down the original rate and multiply by the conversion rate you wish to convert into. • Remember to put the rates you want to cancel out opposite to the original position.

27. Application • In preparation for a trip to Myanmar, Ms. Wong exchanges 12,500 TWD(Taiwan dollars) for USD. Upon arriving in Myanmar, she exchanges the USD for MMK (Myanmar Kyat). How much Kyat will she receive? • Note: The exchange rate at the time were 1 USD = 29 TWD = 870 MMK.

28. Answer 870 1 1 29 1 870 29 1 375,000

29. Practice • 6ft to cm • 20 m/s to km/h

30. Answer • 6ft = 182.88 cm • 20 m/s = 72 km/h

31. Absolute Value Equations • What are absolute values? • How to solve absolute value equations? • Application of absolute value equations.

32. Absolute Vales • Definition: absolute value of a number is its distance from 0 on a number line. • Basic Meaning: Whenever you see a number with the result is always positive since distance can’t be negative. • Tip: absolute values can’t be negative so if an equation shows this , there are no solutions.

33. Solving Absolute Value Equations • Simplify and isolate the • Take off the absolute value symbol and just add a at the beginning of equation. • Ex.

34. Application • The specifications for machining a metal bolt state that it must be 6 cmlong, with a 0.1 cm tolerance. What is the longest the bolt can be? What is the shortest length? • Write an absolute value equation to represent the shortest and longest acceptable lengths.

35. Answers • The longest acceptable length is 6.1 cm and smallest is 6.9 cm.

36. Practice • 3 | x + 1 | - 11 = 4 • 5 - 3 | 2x - 6 | = -7 • 9 + 4 | x - 3 | = 1

37. Answers • 1. • 2. • 3. no solutions (plug back the answer to check)