6.1 Order of Operations and Solving Linear Equations 303 An equation that is true for all real numbers, like the equation in Example 13, is called an identity. An identity has the solution all real numbers. The following table summarizes information regarding contradictions and identities. Type of Equation When Equation Is True How to Write Answer Contradiction Never No solution Identity Always All real numbers Solve Problems Involving Financial Literacy Throughout this textbook we solve a wide variety of real-life application problems that involve money. Included among these applications are calculating sales tax on purchases, determining the interest earned on investments, making monthly car loan payments, purchasing a home with a mortgage, and saving for retirement. Knowledge of these and other financial applications contribute to our financial literacy. Financial literacy refers to the ability to make smart, informed decisions regarding the management of money. Throughout the rest of this textbook, in some exercises in which we are solving financial application problems, we will mark the exercises with a dollar sign icon $ . Our next example corresponds to one such exercise. Note that both sides of the equation are the same. Every real number will satisfy this equation. The solution to this equation is all real numbers. This equation has an infinite number of solutions. 7 Now try Exercise 73 Example 14 Calculating Sales Tax If the sales tax rate on an item is 7%, the sales tax, t, on an item costing x dollars can be determined by the equation = t x 0.07 . Marcelo purchased a new winter coat for $143 and paid 7% sales tax. a) Determine the amount of sales tax Marcelo paid when purchasing the coat. b) Determine the total amount, including sales tax, that Marcelo paid for the coat. c) Marcelo also purchased a pair of gloves and paid 7% sales tax. If the sales tax paid on the gloves was $2.73, determine the price of the gloves prior to the sales tax. Solution a) The coat cost $143, thus = x 143, and the sales tax on the coat is t x t 0.07 0.07(143) 10.01 = = = Thus, the sales tax on the coat is $10.01. b) The coat cost $143 and the sales tax was $10.01. Thus, the total amount paid for the coat, including sales tax is + = $143 10.01 $153.01. c) The sales tax paid on the gloves is $2.73, thus t 2.73. = The cost of the gloves, x, can be determined by solving the following equation: t x x x x 0.07 2.73 0.07 2.73 0.07 0.07 0.07 39 = = = = Thus, the cost of the gloves, before sales tax, is $39. 7 Now try Exercise 75 Instructor Resources for Section 6.1 in MyLab Math • Objective-Level Videos 6.1 • Interactive Concept Video: Order of Operations • PowerPoint Lecture Slides 6.1 • MyLab Exercises and Assignments 6.1
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