2.3. section. hint. helpful. Identities
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1 . Identities, Conditional Equations, and Inconsistent Equations (-15) 79 m m ft 1 m ft FIGURE FOR EXERCISE 95 FIGURE FOR EXERCISE Perimeter of a triangle. If a triangle has sides of length, 1, and meters and a perimeter of 1 meters, then the value of can be found by solving ( 1) ( ) 1. Find the values of, 1, and. m, 4 m, 5 m 97. Cost of a car. Jane paid 9% sales ta and a $150 title and license fee when she bought her new Saturn for a total of $16, If represents the price of the car, then satisfies , Find the price of the car by solving the equation. $14, Cost of labor. An electrician charged Eunice $9.96 for a service call plus $9.96 per hour for a total of $169.8 for installing her electric dryer. If n represents the number of hours for labor, then n satisfies 9.96n Find n by solving this equation..5 hrs In this section. IDENTITIES, CONDITIONAL EQUATIONS, AND INCONSISTENT EQUATIONS In this section, we will solve more equations of the type that we solved in Sections.1 and.. However, some equations in this section have infinitely many solutions, and some have no solution. Identities Conditional Equations Inconsistent Equations Equations Involving Fractions Equations Involving Decimals Simplifying the Process Identities It is easy to find equations that are satisfied by any real number that we choose as a replacement for the variable. For eample, the equations 1,, and 1 1 are satisfied by all real numbers. The equation 5 5 is satisfied by any real number ecept 0 because division by 0 is undefined. Why can t we divide by 0? If 7 0 c, then c 0 7. But c 0 0 no matter what c is.if 0 0 c,then c 0 0,but now c could be any number. Because of these problems we rule out division by 0 and say that division by 0 is undefined. Identity An equation that is satisfied by every real number for which both sides are defined is called an identity. We cannot recognize that the equation in the net eample is an identity until we have simplified each side.
2 80 (-16) Chapter Linear Equations and Inequalities in One Variable E X A M P L E 1 Solving an identity Solve 7 5( 6) 4 ( 5) 8. We first use the distributive property to remove the parentheses: 7 5( 6) 4 ( 5) Combine like terms. This last equation is true for any value of because the two sides are identical. So all real numbers satisfy the original equation, and it is an identity. CAUTION If you get an equation in which both sides are identical, as in Eample 1, there is no need to continue to simplify the equation. If you do continue, you will eventually get 0 0, from which you can still conclude that the equation is an identity. study tip What s on the final eam? Chances are that if your instructor thinks a question is important enough for a test or quiz, that question is also important enough for the final eam. So keep all tests and quizzes, and make sure that you have corrected any mistakes on them.to study for the final eam, write the old questions/problems on note cards, one to a card. Shuffle the note cards and see if you can answer the questions or solve the problems in a random order. Conditional Equations The statement 4 10 is true only on condition that we choose. The equation 4 is satisfied only if we choose or. These equations are called conditional equations. Conditional Equation A conditional equation is an equation that is satisfied by at least one real number but is not an identity. Every equation that we solved in Sections.1 and. is a conditional equation. Inconsistent Equations It is easy to find equations that are false no matter what number we use to replace the variable. Consider the equation 1. If we replace by, we get 1, which is false. If we replace by 4, we get 4 4 1, which is also false. Clearly, there is no number that will satisfy 1. Other eamples of equations with no solutions include, 5, and Inconsistent Equation An equation that has no solution is called an inconsistent equation. E X A M P L E Solving an inconsistent equation Solve ( 4) 4( 7) 7.
3 . Identities, Conditional Equations, and Inconsistent Equations (-17) 81 Use the distributive property to remove the parentheses: ( 4) 4( 7) 7 The original equation Distributive property 14 8 Combine like terms on each side Add to each side Simplify. This last equation is not true for any choice of. So there is no solution to the original equation, and the equation is inconsistent. Keep the following points in mind in solving equations. Summary: Identities and Inconsistent Equations 1. An equation that is equivalent to an equation in which both sides are identical is an identity. The equation is satisfied by all real numbers for which both sides are defined.. An equation that is equivalent to an equation that is always false is inconsistent. The equation has no solution. Equations Involving Fractions We solved some equations involving fractions in Sections.1 and.. Here, we will solve equations with fractions by eliminating all fractions in the first step. All of the fractions will be eliminated if we multiply each side by the least common denominator. E X A M P L E Note that the fractions in Eample will be eliminated if you multiply each side of the equation by any number divisible by both and. For eample, multiplying by 4 yields 1y 4 8y 4 4y 48 y 1. Multiplying by the least common denominator y y Solve 1 1 The least common denominator (LCD) for the denominators and is 6. Since both and divide into 6 evenly, multiplying each side by 6 will eliminate the fractions: y y Multiply each side by 6. y y Distributive property y y 6 y 6 Simplify: 6 y y y 1 Add 6 to each side. y 1 Subtract y from each side. Check 1 in the original equation: Since 1 satisfies the original equation, the solution is 1.
4 8 (-18) Chapter Linear Equations and Inequalities in One Variable Equations involving fractions are usually easier to solve if we first multiply each side by the LCD of the fractions. Equations Involving Decimals When an equation involves decimal numbers, we can work with the decimal numbers or we can eliminate all of the decimal numbers by multiplying both sides by 10, or 100, or 1000, and so on. Multiplying a decimal number by 10 moves the decimal point one place to the right. Multiplying by 100 moves the decimal point two places to the right, and so on. E X A M P L E 4 After you have used one of the properties of equality on each side of an equation, be sure to simplify all epressions as much as possible before using another property of equality. This step is like making sure that all of the injured football players are removed from the field before proceeding to the net play. E X A M P L E 5 Solve the equations in Eamples 4 and 5 without multiplying each side by 10 or 100. It is always good to do a problem by another method. If you can perform operations with decimals, you can solve these equations much faster if you don t remove the decimals. An equation involving decimals Solve 0.p The largest number of decimal places appearing in the decimal numbers of the equation is two (in the number 8.04). Therefore we multiply each side of the equation by 100 because multiplying by 100 moves decimal points two places to the right: 0.p Original equation 100(0.p 8.04) 100(1.6) 100(0.p) 100(8.04) 100(1.6) 0p p p 456 0p Multiplication property of equality Distributive property Subtract 804 from each side. Divide each side by 0. p 15. You can use a calculator to check that 0.(15.) The solution is 15.. Another equation with decimals Solve ( 0) 1.4 First use the distributive property to remove the parentheses: ( 0) 1.4 Original equation Distributive property 10( ) 10(1.4) Multiply each side by Simplify Combine like terms Subtract 80 from each side Simplify. 6 Divide each side by 9.
5 . Identities, Conditional Equations, and Inconsistent Equations (-19) 8 Check 6 in the original equation: 0.5(6) 0.4(6 0) 1.4 Replace by (6) Since both sides of the equation have the same value, 6 is the solution. CAUTION If you multiply each side by 10 in Eample 5 before using the distributive property, be careful how you handle the terms in parentheses: ( 0) ( 0) 14 It is not correct to multiply 0.4 by 10 and also to multiply 0 by 10. Simplifying the Process It is very important to develop the skill of solving equations in a systematic way, writing down every step as we have been doing. As you become more skilled at solving equations, you will probably want to simplify the process a bit. One way to simplify the process is by writing only the result of performing an operation on each side. Another way is to isolate the variable on the side where the variable has the larger coefficient, when the variable occurs on both sides. We use these ideas in the net eample and in future eamples in this tet. E X A M P L E 6 study tip When studying for an eam, start by working the eercises in the Chapter Review. If you find eercises that you cannot do, then go back to the section where the appropriate concepts were introduced. Study the appropriate eamples in the section and work some problems. Then go back to the Chapter Review and continue. WARM-UPS Simplifying the process Solve each equation. a) a 0 b) k 5 k 1 a) Add to each side, then divide each side by : a 0 a Add to each side. a Divide each side by. Check that satisfies the original equation. The solution is. b) For this equation we can get a single k on the right by subtracting k from each side. (If we subtract k from each side, we get k, and then we need another step.) k 5 k 1 5 k 1 Subtract k from each side. 4 k Subtract 1 from each side. Check that 4 satisfies the original equation. The solution is 4. True or false? Eplain your answer. 1. The equation 99 has no solution. True. The equation n n 5n is an identity. True. The equation y y 4y is inconsistent. False
6 84 (-0) Chapter Linear Equations and Inequalities in One Variable WARM-UPS (continued) 4. All real numbers satisfy the equation 1 1. False 5. The equation 5a 0 is an inconsistent equation. False 6. The equation t t is a conditional equation. True 7. The equation w 0.1w 0.9w is an identity. True 8. The equation is equivalent to 0 8. False 9. The equation 1 is an identity. True 10. The solution to h 8 0is 8. True. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What is an identity? An identity is an equation that is satisfied by all numbers for which both sides are defined.. What is a conditional equation? A conditional equation has at least one solution but is not an identity.. What is an inconsistent equation? An inconsistent equation has no solutions. 4. What is the usual first step when solving an equation involving fractions? If an equation involves fractions we usually multiply each side by the LCD of all of the fractions. 5. What is a good first step for solving an equation involving decimals? If an equation involves decimals we usually multiply each side by a power of 10 to eliminate all decimals. 6. Where should the variable be when you are finished solving an equation? The goal is to get the variable isolated on one side of the equation. Solve each equation. Identify each as a conditional equation, an inconsistent equation, or an identity. See Eamples 1 and All real numbers, identity All real numbers, identity 9. a 1 a r 7 r No solution, inconsistent No solution, inconsistent 11. y 4y 1y 1. 9t 8t 7 0, conditional 7, conditional 1. 4 (w 1) w (w ) 1 No solution, inconsistent (w ) (w 1) 7w 4 All real numbers, identity 15. (m 1) (m ) No solution, inconsistent 16. 5(m 1) 6(m ) 4 m No solution, inconsistent 17. 1, conditional , conditional 19. (5 ) No solution, inconsistent 0. (5 ) 0 4, conditional 1. ( )(5 z) 0 All real numbers, identity. ( 4 8)p 0 All real numbers, identity. 0 0 All nonzero real numbers, identity 4. All real numbers, identity 5. All real numbers, identity 6. 1 All nonzero real numbers, identity Solve each equation by first eliminating the fractions. See Eample w w a a z z m m p 5 4 p q 6 5 q v 1 v k 5 6 k Solve each equation by first eliminating the decimal numbers. See Eamples 4 and a 0. 0.a 8. 80
7 . Identities, Conditional Equations, and Inconsistent Equations (-1) b.4 0.b r 0.4r t t y 0.0(y 50) y 0.08(y 100) ( 00) ( 100) 5 00 Solve each equation. If you feel proficient enough, try simplifying the process, as described in Eample z s c b t y q q p 4p Solve each equation ( 4) ( ) 1 All real numbers 7. u (u 4) 4(u 5) No solution 7. 5( n) 4(n ) 9n All real numbers 74. 4(t 5) (t ) a t 79. y y w w ( 4) ( 100) , ( 98) Solve each problem. 87. Sales commission. Danielle sold her house through an agent who charged 8% of the selling price. After the commission was paid, Danielle received $117,760. If is the selling price, then satisfies ,760. Solve this equation to find the selling price. $18, Raising rabbits. Before Roland sold two female rabbits, half of his rabbits were female. After the sale, only onethird of his rabbits were female. If represents his original number of rabbits, then 1 1 ( ). Solve this equation to find the number of rabbits that he had before the sale. 8 rabbits 89. Eavesdropping. Reginald overheard his boss complaining that his federal income ta for 1997 was $4,76. a) Use the accompanying graph to estimate his boss s taable income for $150,000 b) Find his boss s eact taable income for 1997 by solving the equation,5 0.1( 99,600) 4,76. $17,48.87 Federal income ta (thousands of dollars) Married filing jointly Taable income (thousands of dollars) FIGURE FOR EXERCISE Federal taes. According to Bruce Harrell, CPA, the federal income ta for a class C corporation is found by solving a linear equation. The reason for the equation is that the amount of federal ta is deducted before the state ta is figured, and the amount of state ta is deducted before the federal ta is figured. To find the amount of federal ta for a corporation with a taable income of $00,000, for which the federal ta rate is 5% and the state ta rate is 10%, Bruce must solve 0.5[00, (00,000 )]. Solve the equation for Bruce. $46,15.85
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