Lesson 12: Solving Equations

Transcription

1 Exploratory Exercises 1. Alonzo was correct when he said the following equations had the same solution set. Discuss with your partner why Alonzo was correct. (xx 1)(xx + 3) = 17 + xx (xx 1)(xx + 3) = xx He then said that (xx 1)(xx + 3) = 17 + xx and (xx 1)(xx + 3) = xx should have the same solution set. Is he correct? Explain your reasoning. 3. Finally, Alonzo said the equations (xx 1)(xx + 3) = 17 + xx and 3(xx 1)(xx + 3) = xx should have the same solution set. What do you think? Why? S.103

2 4. Consider the equation xx = 7 xx. a. Verify that this has the solution set { 3, 2}. Draw this solution set as a graph on the number line. We will later learn how to show that these happen to be the ONLY solutions to this equation. b. Let s add 4 to both sides of the equation and consider the new equation xx = 11 xx. Verify 2 and 3 are still solutions. c. Let s now add xx to both sides of the equation and consider the new equation xx xx = 11. Are 2 and 3 still solutions? d. Let s add 5 to both sides of the equation and consider the new equation xx 2 + xx = 6. Are 2 and 3 still solutions? e. Let s go back to part (d) and add 3xx 3 to both sides of the equation and consider the new equation xx 2 + xx + 3xx 3 = 6 + 3xx 3. Are 2 and 3 still solutions? S.104

3 From Exercise 4, whenever aa = bb is true, then aa + cc = bb + cc will also be true for all real numbers cc. 5. What if aa = bb is false? Will aa + cc = bb + cc will also be false? 6. Is it also okay to subtract a number from both sides of the equation? Explain your reasoning. 7. Let s go back to Exercise 4 and this time multiply both sides by 1 6 to get xx2 +xx solutions? 6 = 1. Are 2 and 3 still Whenever aa = bb is true, then aacc = bbcc will also be true, and whenever aa = bb is false, aacc = bbcc will also be false for all nonzero real numbers cc. So, we have said earlier that applying the distributive, associative, and commutative properties does not change the solution set, and now we see that applying the additive and multiplicative properties of equality also preserves the solution set (does not change it). 8. Is xx = 5 an equation? If so, what is its solution set? This example is so simple that it may be hard to wrap your brain around, but it points out that if ever we have an equation that is this simple, we know its solution set. 9. Determine each solution set. (a) ww 2 = 64 (b) 7 + PP = 5 (c) 3ββ = 10 S.105

4 Here s the strategy: If we are faced with the task of solving an equation, that is, finding the solution set of the equation: Use the commutative, associative, distributive properties AND Use the properties of equality (adding, subtracting, multiplying by non-zeros, dividing by non-zeros) to keep rewriting the equation into one whose solution set you easily recognize. (We observed that the solution set will not change under these operations.) This usually means rewriting the equation so that all the terms with the variable appear on one side of the equation. 10. To illustrate this idea, in this exercise you and your team will solve the equation 3xx + 4 = 8xx 16 starting in four different ways. Determine who will be Student 1, 2, 3 and 4. Then solve for xx using the given starting point. 3xx + 4 = 8xx 16 Student 1 Student 2 Student 3 Student 4 Subtract 3xx from both Subtract 4 from both Subtract 8xx from both Add 16 to both sides sides sides sides 11. Did everyone get the same value for x? If not, check to see where a mistake was made. Remember you must do the same thing to both sides of the equation to keep it balanced. S.106

5 Lesson Summary If xx is a solution to an equation, it will also be a solution to the new equation formed when the same number is added to (or subtracted from) each side of the original equation or when the two sides of the original equation are multiplied by (or divided by) the same nonzero number. These are referred to as the properties of equality. If you are faced with the task of solving an equation, that is, finding the solution set of the equation: Use the commutative, associative, and distributive properties, AND use the properties of equality (adding, subtracting, multiplying by nonzeros, dividing by nonzeros) to keep rewriting the equation into one whose solution set you easily recognize. Homework Problem Set Solve the following equations, check your solutions, and then graph the solution sets vv = 2(8vv 7) 2. 2(6bb + 8) = 4 + 6bb 3. (xx 2)(xx 2) = x xx = 7(1 + 7xx) nn = 8(3 + 4nn) + 3nn 6. (xx 1)(xx + 5) = xx 2 + 4xx 2 S.107

6 7. xx 2 7 = xx 2 6xx aa + 5aa = 3aa 5aa xx = 1 5xx + 2xx 10. 4(xx 2) = 8(xx 3) (1 nn) = 6 6nn aa = 5(aa + 6) pp = 6pp + 5(pp + 3) 14. xx xx+2 = xx 9 = xx ( 5xx 6) = 22 xx xx = xx (2rr 0.3) + 0.5(4rr + 3) = 64 S.108