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Solving Linear Equations

Solving Linear Equations. Algebra—Chapter 3. Section 3.1. Solving Equations Using Addition & Subtraction. Equivalent equations: Equations with the same solutions Apply transformations to create equivalent equations Goal: apply transformations to isolate the variable.

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Solving Linear Equations

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  1. Solving Linear Equations Algebra—Chapter 3

  2. Section 3.1 • Solving Equations Using Addition & Subtraction

  3. Equivalent equations: • Equations with the same solutions • Apply transformations to create equivalent equations • Goal: apply transformations to isolate the variable

  4. Transformations that we can use: • Add same number to each side • Subtract same number from each side • Simplify one or both sides • Interchange the sides • “Balance the scale”

  5. More definitions! • Inverse operations: • Operations that undo each other • Linear equation:

  6. Solving equations • x – 5 = – 13 • – 8 = n – (-6)

  7. Solving equations • 11 = r – 4 • t – 2 = 6 • 19 – (-y) = 25

  8. Model a real-life problem • Your bank balance is $42. If you write a check to buy a pair of shoes, your balance would be – $5. • Use an equation to model the cost of the shoes.

  9. Assignment • Section 3.1 • Page 135 • 21-44 All

  10. Section 3.2 • Solving Equations Using Multiplication and Division

  11. More transformations that we can use: • Multiply each side by same non-zero number • Divide each side by same non-zero number

  12. Examples:

  13. Examples

  14. Properties of equality • Addition property • If a = b, then a + c = b + c • Subtraction Property • If a = b, then a - c = b - c • Multiplication Property • If a = b, then a • c = b • c • Division Property • If a = b, and c ≠ 0, then a ÷ c = b ÷ c

  15. Ratios: • If two quantities are measured in the same units, then the ratio of a to b is

  16. Similar triangles: • Have equal corresponding angles • Ratio of lengths of corresponding sides are equal B E D F A C

  17. Assignment • Section 3.2 • Page 141-143 • 14-46 even, 48-53 all, 58, 59

  18. Section 3.3 • Solving Multi-step Equations

  19. Multi-step equations • Involve 2 or more transformations • Simplify one or both sides (if necessary) • Use inverse operations to isolate variable

  20. Solve:

  21. Solve:

  22. Assignment: • Section 3.3 • Page 148 • 16-40

  23. Multi-step equations—Day 2

  24. Consecutive integers • The sum of two consecutive integers is 25. What are the two integers?

  25. Using formulas • Use the formula to convert 102 degrees Fahrenheit to Celsius

  26. Assignment • Section 3.3 • P 149-150 • 46-65

  27. Section 3.4 • Solving Equations with Variables on Both Sides

  28. Collect all the variables on one side • Either left or right, no difference

  29. 6x + 22 = -3x + 31 • 64 – 12w = 6w

  30. Many solutions—or—no solution • Identity: • Examples: 3(x+2) = 3x + 6 x + 2 = x + 4

  31. Solve:

  32. Assignment • Section 3.4 • Page 157 • 12-34 all

  33. Solving Equations with Variables on Both Sides—Day 2

  34. Member-ship fee Visits to the gym Cost per visit Member-ship fee Visits to the gym Cost per visit + = + * * • A gym offers two packages for yearly membership. The first plan costs $50 to be a member. Then each visit to the gym is $5. The second plan costs $200 for a membership fee plus $2 per visit. Which membership is a better deal?

  35. Unit Analysis: • Check the units to make sure your units match

  36. Find resulting unit of measure • (Dollars per hour) * (hour) • (years)*(people per year)

  37. Assignment • Section 3.4 • Practice A

  38. Section 3.5 • Linear Equations and Problem Solving

  39. Problem solving strategies • Draw a diagram • Use a table/graph as a check

  40. Draw a Diagram • A page of pictures for a yearbook is 8.5 inches by 11 inches. • The top margin is .75 inches, and the bottom is 1.25 inches. • The space between the pictures is 3/16 inch. • How high can each picture be to fit seven down the length of the page?

  41. Use a table to check • At West High School, 362 students take Spanish. • This number has been increasing at a rate of 20 per year. • The number of students taking French is 259 and has been decreasing at a rate of about 3 per year. • At these rates, when will there be two times as many students taking Spanish as taking French?

  42. Using a graph as a check • Horizontal axis • Vertical axis

  43. Jim can run 11 ft/sec and Sarah can run 13 ft/sec. How far ahead of Sarah must Jim be to fall behind Sarah in the first 15 seconds that they run?

  44. Assignment • Section 3.5 • Page 163-165 • 6-21 all, 31-39 odd

  45. Section 3.6 • Solving Decimal equations

  46. Round-off error: • Exact answer isn’t always practical • Example: Restaurant bill is 12.95, want to split evenly 3 ways. • 3x=12.95

  47. If four people are sharing the cost of a monthly phone bill of $58.25, what is each person’s share of the bill?

  48. Equations with decimals: • Same methods as without decimals • Round answer (usually to nearest hundredth)

  49. Example: • Round to nearest hundredth 9.92x – 6.13 = 5.96 – 7.28x

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