1.1. Variables in Algebra. What you should learn

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1 1.1 Variables in Algebra Wat you sould learn GOAL 1 Evaluate a variable GOAL 2 Write a variable expression tat models a real-life situation, as in te iking problem in Example 5. Wy you sould learn it To express real-life number relationsips, suc as pitcing speed in Exs. 48 and 49. GOAL 1 EVALUATING VARIABLE EXPRESSIONS A variable is a letter tat is used to represent one or more numbers. Te numbers are te values of te variable. A variable expression is a collection of numbers, variables, and operations. Here are some examples. VARIABLE EXPRESSION MEANING OPERATION 8y 8 y 8(y) 8 times y Multiplication 1 6 b 16 b 16 divided by b Division 4 + s 4 plus s Addition 9 º x 9 minus x Subtraction Te expression 8y is usually not written as 8 ª y because of possible confusion wit te variable x. Replacing eac variable in an expression by a number is called evaluating te Te resulting number is te value of te Write te variable Substitute values for variables. Simplify te numerical EXAMPLE 1 Evaluating a Variable Expression Evaluate te expression wen y = 2. a. 8y b. 1 y0 c. y + 3 d. 14 º y a. 8y = 8(2) Substitute 2 for y. = 16 Simplify. b = y 2 Substitute 2 for y. = 5 Simplify. c. y + 3 = Substitute 2 for y. = 5 Simplify. d. 14 º y = 14 º 2 Substitute 2 for y. = 12 Simplify. 1.1 Variables in Algebra 3

2 Travel EXAMPLE 2 Evaluating a Real-Life Expression Average speed is given by te following formula. Average speed = D istance = d Time t Find te average speed (in miles per our) of a car tat traveled 180 miles from Boise, Idao, to te Minidoka National Wildlife Refuge in 3 ours. Average speed = d t = 18 0 Substitute 180 for d and 3 for t. 3 = 60 Simplify. Te average speed was 60 miles per our. EXAMPLE 3 Evaluating a Geometric Expression GEOMETRY CONNECTION Te perimeter of a triangle is equal to te sum of te lengts of its sides: a + b + c. Find te perimeter of te triangle. Te dimensions are in feet. a = 8 c = 17 b = 15 Perimeter = a + b + c = Substitute values. = 40 Simplify. Te triangle as a perimeter of 40 feet. EXAMPLE 4 Evaluating Simple Interest Interest Te simple interest earned by money P (called te principal) at an annual interest rate r for t years is given by Prt. You deposit $650 at a rate of 8% per year. How muc simple interest will you earn after one alf of a year? Skills Review For elp wit writing percents as decimals, see pp Simple interest = Prt = (650)(0.08)(0.5) Substitute 650 for P, 0.08 for r, and 0.5 for t. = 26 Simplify. After one alf of a year, you will ave $26 of simple interest. 4 Capter 1 Connections to Algebra

3 FOCUS ON APPLICATIONS GOAL 2 MODELING A - SITUATION Writing te units of eac variable in a real-life problem elps you determine te units for te answer. Tis is called unit analysis and it is often used in problem solving in science. Wen te same units of measure occur in te numerator and te denominator of an expression, you can cancel te units. In real-life problems, you may need to translate words into matematics. One way to do tis is to use te verbal model sown in Example 5. EXAMPLE 5 Finding Time COASTAL REDWOODS, found in California and Oregon, are te tallest trees on Eart. Only 4% of original redwood forests remains. Te tallest living redwood is 370 feet ig. HIKING AMONG REDWOODS You plan to go iking in te Jededia Smit Redwoods State Park in California. You estimate you ll ike at a rate of 2 miles per our on a steep trail. How long will it take you to ike from Howland Hill Road along te Boy Scout Tree Trail and back? JEDEDIAH SMITH REDWOODS STATE PARK Boy Scout Tree Trail Hiouci Information Center Little Bald Hills Trail Smit River Howland Hill Road Round Trip Distances Boy Scout Tree Trail Little Bald Hills Trail 7.4 miles 9 miles Redwood National Park Boundary Trail Paved Road Source: Redwood National and State Parks PROBLEM SOLVING STRATEGY Te map sows tat te round-trip distance is about 7.4 miles. VERBAL MODEL = Di stance Time Rate LABELS Time = t Distance = 7.4 Rate = 2 (ours) (miles) (miles per our) ALGEBRAIC MODEL t = 7.4 Write algebraic model. 2 = 3.7 Simplify. It sould take you about 3.7 ours to ike te Boy Scout Tree Trail. UNIT ANALYSIS Use unit analysis to ceck tat ours are te units of te solution. Time = Di stance mi = = mi = Rate m i/ m i 1.1 Variables in Algebra 5

4 GUIDED PRACTICE Vocabulary Ceck Concept Ceck 1. Explain wat it means to evaluate a variable 2. Wat operation is indicated by te expression? a. 4y b. 7 c. t + 8 d. 3 º t d Skill Ceck 3. Write a variable expression for 5 divided by r. 4. How is unit analysis elpful in solving real-life problems? Evaluate te expression wen y = y 6. 2 y4 7. y y º 2 9. y º y y y Simplify te real-life expression and sow unit analysis. 15 mi 13. time = 14. perimeter = 3 cm + 4 cm + 5 cm 5 mi/ distance = (60 mi/)(2.3 ) 16. simple interest = ($100) (2.5 years) year HIKING In Exercises 17 and 18, you want to ike a round-trip distance of 10 miles from te Hiouci Information Center along te Little Bald Hills Trail and back. Calculate ow long it will take if you ike at a rate of 1.25 miles per our. 17. Write a verbal model, provide labels, and write an algebraic model. 18. Sow unit analysis. PRACTICE AND APPLICATIONS Extra Practice to elp you master skills is on p HOMEWORK HELP Example 1: Exs Example 2: Exs Example 3: Ex. 32 Example 4: Exs. 31, 33, 34 Example 5: Ex. 39 EVALUATING EXPRESSIONS Evaluate te expression for te given value of te variable x wen x = b º 7 wen b = d wen d = p wen p = r(10) wen r = a wen a = wen x = x 3 x wen x = d wen d = wen t = t 8 º p wen p = t wen t = INTEREST EARNED You invest $80 at a simple annual interest rate of 2%. How muc simple interest would you earn in 1.5 years? Use unit analysis to ceck te units in your response. 32. GEOMETRY CONNECTION Te perimeter of a square is equal to 4s, were s is te lengt of one side. Find te perimeter of te square at te rigt. Sow unit analysis. 7 ft 6 Capter 1 Connections to Algebra

5 ERROR ANALYSIS In Exercises 33 and 34, Samanta deposits $300 in an account tat earns an annual interest rate of 2.5%. After 9 monts, se computes te simple interest Wat mistake did Samanta make? ($300) y e (0.75 year) = $ ar 34. Wat is te correct amount of simple interest earned after 9 monts? HOMEWORK HELP Visit our Web site for elp wit Exs INTERNET FOCUS ON PEOPLE CALCULATING AVERAGE SPEEDS In Exercises 35 38, find te average speed for te given distance and time. Sow unit analysis to ceck units. 35. A train travels 75 miles in 55 minutes. 36. In 5 seconds, an atlete runs 40 feet. 37. A orse gallops 4 kilometers in 30 minutes. 38. Dick Rutan and Jeana Yeager flew nonstop around te world, a distance of 24, miles, in ours. Source: National Air and Space Museum 39. DRIVING TIME If you are driving at a constant speed of 96 kilometers per our, ow long will it take you to travel 288 kilometers? CALORIES BURNED In Exercises 40 42, use te following information. Te number of calories burned wile doing an activity can be expressed by rm, were r is te rate of calories burned and m is te number of minutes spent doing te activity. Source: Home & Garden Bulletin No. 72, U.S. Government Printing Office 40. A 120-pound student playing volleyball burns 2.7 calories per minute. If te student plays for 30 minutes, ow many calories does te student burn? 41. An in-line skater, wo is te same weigt as te student in Exercise 40, burns 387 calories in 90 minutes. How many calories does te in-line skater burn per minute? 42. Wic activity burns more calories per minute, volleyball or in-line skating? 43. MOUNTAIN CLIMBING Heidi Zimmer plans to climb te igest peak in eac continent. Se as already climbed summits in Nort America, Europe, Africa, and Sout America. Copy and complete te table. Convert, te eigt in meters, to te eigt in feet by dividing eac given value of by Te last column sows ow muc sorter eac mountain is tan Mt. Everest, wic is 29,029 feet ig. HEIDI ZIMMER is te first deaf woman to climb Mt. McKinley. It took er 18 days to reac te top of Mt. McKinley. APPLICATION LINK INTERNET Mountain, Continent }} ,029 }} Mt. McKinley, Nort America 6194?? Mt. Elbrus, Europe 5633?? Mt. Kilimanjaro, Africa 5963?? Mt. Aconcagua, Sout America 6959?? Source: Peakware World Mountain Encyclopedia 1.1 Variables in Algebra 7