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SOLVING EQUATIONS AND PROBLEMS

SOLVING EQUATIONS AND PROBLEMS. CHAPTER 3. Section 3-1 Transforming Equations: Addition and Subtraction. Addition Property of Equality. If a, b, and c are any real numbers, and a = b, then a + c = b + c and c + a = c + b. Subtraction Property of Equality.

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SOLVING EQUATIONS AND PROBLEMS

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  1. SOLVING EQUATIONS AND PROBLEMS CHAPTER 3

  2. Section 3-1 Transforming Equations: Addition and Subtraction

  3. Addition Property of Equality If a, b, and c are any real numbers, and a = b, then a + c = b + c and c + a = c + b

  4. Subtraction Property of Equality If a, b, and c are any real numbers, and a = b, then a - c = b - c and c - a = c - b

  5. Equivalent Equations Equations having the same solution set over a given domain. -5 = n + 13 and -18 = n are equivalent

  6. Transforming an Equation into an Equivalent Equation

  7. Transformation by Substitution Substitute an equivalent expression for any expression in a given equation.

  8. Transformation by Addition Add the same real number to each side of a given equation.

  9. Transformation by Subtraction Subtract the same real number from each side of a given equation.

  10. EXAMPLES Solve: x – 8 = 17 Add8 x – 8 + 8 = 17 + 8 x = 25

  11. EXAMPLES Solve: -5 = n + 13 Subtract 13 -5 -13 = n + 13 – 13 -18 = n

  12. EXAMPLES Solve: x + 5 = 9 Subtract 5 x + 5 – 5 = 9 - 5 x = 4

  13. Section 3-2 Transforming Equations: Multiplication and Division

  14. Multiplication Property of Equality If a, b, and c are any real numbers, and a = b, then ca = cb and ac = bc

  15. Division Property of Equality If a and b are real numbers, c is any nonzero real number, and a = b, then a/c = b/c

  16. Transformation by Multiplication Multiply each side of a given equation by the same nonzero real number.

  17. Transformation by Division Divide each side of a given equation by the same nonzero real number.

  18. EXAMPLES Solve: • 6x = 222 • 8 = -2/3t • m/3 = -5

  19. Section 3-3 Using Several Transformations

  20. Inverse Operations For all real numbers a and b, (a + b) – b = a and (a – b) + b = a

  21. Inverse Operations For all real numbers a and all nonzero real numbers b (ab)  b = a and (a  b)b = a

  22. EXAMPLES Solve: • 5n – 9 = 71 • 1/5x + 2 = -1 • 40 = 2x + 3x • 8(w + 1) – 3 = 48

  23. 3-4 Using Equations to Solve Problems

  24. EXAMPLES The sum of 38 and twice a number is 124. Find the number.

  25. EXAMPLES The perimeter of a trapezoid is 90 cm. The parallel bases are 24 cm and 38 cm long. The lengths of the other two sides are consecutive odd integers. What are the lengths of these other two sides?

  26. Solution 38 x x + 2 24

  27. 3-5 Equations with Variables on Both Sides

  28. EXAMPLES • 6x = 4x + 18 • 3y = 15 – 2y • (4 + y)/5 = y • 3/5x = 4 – 8/5x • 4(r – 9) + 2 = 12r + 14

  29. 3-6 Problem Solving: Using Charts

  30. PROBLEM A swimming pool that is 25 m long is 13 m narrower than a pool that is 50 m long. Organize in chart form.

  31. SOLUTION

  32. PROBLEM A roll of carpet 9 ft wide is 20 ft longer than a roll of carpet 12 ft wide. Organize in chart form.

  33. SOLUTION

  34. PROBLEM An egg scrambled with butter has one more gram of protein than an egg fried in butter. Ten scrambled eggs have as much protein as a dozen fried eggs. How much protein is in one fried egg?

  35. SOLUTION

  36. 3-7 Cost, Income, and Value Problems

  37. Formulas • Cost = # of items x price/item • Income = hrs worked x wage/hour • Total value = # of items x value/item

  38. PROBLEM Tickets for the senior class play cost $6 for adults and $3 for students. A total of 846 tickets worth $3846 were sold. How many student tickets were sold?

  39. SOLUTION

  40. PROBLEM Marlee makes $5 an hour working after school and $6 an hour working on Saturdays. Last week she made $64.50 by working a total of 12 hours. How many hours did she work on Saturday?

  41. SOLUTION

  42. THE END The End

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