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# 3. using the principles together

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### 3. using the principles together

1. 1. 1.3 Using the Principles Together OBJECTIVES a Solve equations using both the addition principle and the multiplication principle. b Solve equations in which like terms may need to be collected. c Solve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 1
2. 2. 1.3 Using the Principles Together Solve equations using both the addition principle and a the multiplication principle. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 2
3. 3. 1.3 Using the Principles Together a EXAMPLE Solve equations using both the addition principle and the multiplication principle. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 3
4. 4. 1.3 Using the Principles Together Solve equations using both the addition principle and a the multiplication principle. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 4
5. 5. 1.3 Using the Principles Together b Solve equations in which like terms may need to be collected. If there are like terms on one side of the equation, we collect them before using the addition principle or the multiplication principle. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 5
6. 6. 1.3 Using the Principles Together Solve equations in which like terms may need to be b collected. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 6
7. 7. 1.3 Using the Principles Together b Solve equations in which like terms may need to be collected. If there are like terms on opposite sides of the equation, we get them on the same side by using the addition principle. Then we collect them. In other words, we get all the terms with a variable on one side of the equation and all the terms without a variable on the other side. If there are like terms on one side at the outset, they should be collected first. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 7
8. 8. 1.3 Using the Principles Together Solve equations in which like terms may need to be b collected. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 8
9. 9. 1.3 Using the Principles Together Solve equations in which like terms may need to be b collected. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 9
10. 10. 1.3 b Using the Principles Together Solve equations in which like terms may need to be collected. In general, equations are easier to solve if they do not contain fractions or decimals. The easiest way to clear an equation of fractions is to multiply every term on both sides by the least common multiple of all the denominators. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 10
11. 11. 1.3 Using the Principles Together Solve equations in which like terms may need to be b collected. EXAMPLE The denominators are 3, 6, and 2. The number 6 is the least common multiple of all the denominators. We multiply by 6 on both sides of the equation. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 11
12. 12. 1.3 b Using the Principles Together Solve equations in which like terms may need to be collected. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 12
13. 13. 1.3 b Using the Principles Together Solve equations in which like terms may need to be collected. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 13
14. 14. 1.3 Using the Principles Together Solve equations in which like terms may need to be b collected. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 14
15. 15. 1.3 Using the Principles Together b Solve equations in which like terms may need to be collected. To clear an equation of decimals, we count the greatest number of decimal places in any one number. If the greatest number of decimal places is 1, we multiply every term on both sides by 10; if it is 2, we multiply by 100; and so on. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 15
16. 16. 1.3 Using the Principles Together Solve equations in which like terms may need to be b collected. EXAMPLE The greatest number of decimal places in any one number is two. Multiplying by 100, which has two 0’s, will clear all decimals. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 16
17. 17. 1.3 Using the Principles Together Solve equations in which like terms may need to be b collected. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 17
18. 18. 1.3 Using the Principles Together c Solve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions. To solve certain kinds of equations that contain parentheses, we first use the distributive laws to remove the parentheses. Then we proceed as before. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 18
19. 19. 1.3 c Using the Principles Together Solve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 19
20. 20. 1.3 Using the Principles Together 1. Multiply on both sides to clear the equation of fractions or decimals. (This is optional, but it can ease computations.) 2. If parentheses occur, multiply to remove them using the distributive laws. 3. Collect like terms on each side, if necessary. 4. Get all terms with variables on one side and all numbers (constant terms) on the other side, using the addition principle. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 20
21. 21. 1.3 Using the Principles Together 5. Collect like terms again, if necessary. 6. Multiply or divide to solve for the variable, using the multiplication principle. 7. Check all possible solutions in the original equation. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 21
22. 22. 1.3 c Using the Principles Together Solve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 22
23. 23. 1.3 c Using the Principles Together Solve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 23
24. 24. 1.3 c Using the Principles Together Solve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 24
25. 25. 1.3 c Using the Principles Together Solve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 25
26. 26. 1.3 Using the Principles Together Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 26
27. 27. 1.3 c Using the Principles Together Solve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions. EXAMPLE Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 27