116 CHAPTER 1 Equations and Inequalities Step 4 Solve. 0.01325x + 11,931 - 0.0291x = 8761 Distributive property 11,931 - 0.01585x = 8761 Combine like terms. -0.01585x = -3170 Subtract 11,931. x = 200,000 Divide by -0.01585. Step 5 State the answer. The artist should invest $200,000 at 2.65% for 6 months and $410,000 - $200,000 = $210,000 at 2.91% for 1 yr to earn $8761 in interest. Step6 Check. The 6-month investment earns $200,00010.0265210.52 = $2650, and the 1-yr investment earns $210,00010.02912112 = $6111. The total amount of interest earned is $2650 + $6111 = $8761, as required. S Now Try Exercise 35. EXAMPLE 4 Solving an Investment Problem An artist has sold a painting for $410,000. He invests a portion of the money for 6 months at 2.65% and the rest for a year at 2.91%. His broker tells him the two investments will earn a total of $8761 in simple interest. How much should be invested at each rate to obtain that amount of interest? SOLUTION Step 1 Read the problem. We must find the amount to be invested at each rate. Step 2 Assign a variable. Let x = the dollar amount to be invested for 6 months at 2.65%. 410,000 - x = the dollar amount to be invested for 1 yr at 2.91%. Step3 Write an equation. The sum of the two interest amounts must equal the total interest earned. + = Interest from 2.65% investment (1111111)1111111* Interest from 2.91% investment (1111111)1111111* Total interest (1)1* 0.026510.52x + 0.02911410,000 - x2 = 8761 P Invested Amount r Interest Rate (%) t Time (in years) I Interest Earned x 2.65 0.5 x10.0265210.52 410,000 - x 2.91 1 1410,000 - x210.02912112 Summarize the information in a table using the formula I = Prt. Modeling with Linear Equations A mathematical model is an equation (or inequality) that describes the relationship between two quantities. A linear model is a linear equation. The next examples apply linear models.
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