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Transcript
Warm-Up Exercises 1. Solve the linear system using substitution. 2x + y = 12 3x – 2y = 11 ANSWER (5, 2) 2. One auto repair shop charges $30 for a diagnosis and $25 per hour for labor. Another auto repair shop charges $35 per hour for labor. For how many hours are the total charges for both of the shops the same? ANSWER 3h EXAMPLE Warm-Up1Exercises Use addition to eliminate a variable Solve the linear system: 2x + 3y = 11 –2x + 5y = 13 Equation 1 Equation 2 SOLUTION STEP 1 STEP 2 Add the equations to eliminate one variable. Solve for y. 2x + 3y = 11 –2x + 5y = 13 8y = 24 y=3 EXAMPLE Warm-Up1Exercises Use addition to eliminate a variable STEP 3 Substitute 3 for y in either equation and solve for x. 2x + 3y = 11 2x + 3(3) = 11 x=1 ANSWER The solution is (1, 3). Write Equation 1 Substitute 3 for y. Solve for x. EXAMPLE Warm-Up1Exercises Use addition to eliminate a variable CHECK Substitute 1 for x and 3 for y in each of the original equations. 2x + 3y = 11 ? 2x + 5y = 13 ? 2(1) + 3(3) = 11 2(1) + 5(3) = 13 11 = 11 13 = 13 EXAMPLE Warm-Up2Exercises Use subtraction to eliminate a variable Solve the linear system: 4x + 3y = 2 5x + 3y = –2 Equation 1 Equation 2 SOLUTION STEP 1 Subtract the equations to eliminate one variable. STEP 2 Solve for x. 4x + 3y = 2 5x + 3y = –2 –x = 4 x = 4 EXAMPLE Warm-Up2Exercises Use subtraction to eliminate a variable STEP 3 Substitute 4 for x in either equation and solve for y. 4x + 3y = 2 Write Equation 1. Substitute –4 for x. 4(–4) + 3y = 2 y=6 Solve for y. ANSWER The solution is (–4, 6). EXAMPLE Warm-Up3Exercises Arrange like terms Solve the linear system: 8x – 4y = –4 4y = 3x + 14 Equation 1 Equation 2 SOLUTION STEP 1 STEP 2 STEP 3 Rewrite Equation 2 so that the like terms are arranged in columns. 8x – 4y = –4 8x – 4y = –4 4y = 3x + 14 3x + 4y = 14 5x = 10 Add the equations. Solve for x. x=2 EXAMPLE Warm-Up3Exercises Arrange like terms STEP 4 Substitute 2 for x in either equation and solve for y. 4y = 3x + 14 4y = 3(2) + 14 y=5 Write Equation 2. Substitute 2 for x. Solve for y. ANSWER The solution is (2, 5). Warm-Up Exercises GUIDED PRACTICE for Example 1,2 and 3 Solve the linear system: 1. 4x – 3y = 5 –2x + 3y = –7 ANSWER (–1, –3) Warm-Up Exercises GUIDED PRACTICE for Example 1,2 and 3 Solve the linear system: 2. – 5x – 6y = 8 5x + 2y = 4 ANSWER (2, –3) Warm-Up Exercises GUIDED PRACTICE for Example 1,2 and 3 Solve the linear system: 3. 6x – 4y = 14 – 3x + 4y = 1 ANSWER (5, 4) Warm-Up Exercises GUIDED PRACTICE for Example 1,2 and 3 Solve the linear system: 4. 7x – 2y = 5 7x – 3y = 4 ANSWER (1, 1) Warm-Up Exercises GUIDED PRACTICE for Example 1,2 and 3 Solve the linear system: 5. 3x + 4y = –6 2y = 3x + 6 ANSWER (–2, 0) Warm-Up Exercises GUIDED PRACTICE for Example 1,2 and 3 Solve the linear system: 6. 2x + 5y = 12 5y = 4x + 6 ANSWER (1, 2) Warm-Up4Exercises EXAMPLE Write and solve a linear system KAYAKING During a kayaking trip, a kayaker travels 12 miles upstream (against the current) and 12 miles downstream (with the current), as shown. The speed of the current remained constant during the trip. Find the average speed of the kayak in still water and the speed of the current. Warm-Up4Exercises EXAMPLE Write and solve a linear system STEP 1 Write a system of equations. First find the speed of the kayak going upstream and the speed of the kayak going downstream. Upstream: d = rt Downstream: d = rt 12 = r 3 12 = r 2 4=r 6=r Warm-Up4Exercises EXAMPLE Write and solve a linear system Use the speeds to write a linear system. Let x be the average speed of the kayak in still water, and let y be the speed of the current. Equation 1: Going upstream x – y = 4 Warm-Up4Exercises EXAMPLE Write and solve a linear system Equation 2: Going downstream x + y = 6 Warm-Up4Exercises EXAMPLE Write and solve a linear system STEP 2 Solve the system of equations. x–y=4 Write Equation 1. x+y=6 Write Equation 2. 2x = 10 x=5 Add equations. Solve for x. Substitute 5 for x in Equation 2 and solve for y. Warm-Up4Exercises EXAMPLE Write and solve a linear system 5+y=6 y=1 Substitute 5 for x in Equation 2. Subtract 5 from each side. Warm-Up Exercises GUIDED PRACTICE 7. for Example 4 WHAT IF? In Example 4, suppose it takes the kayaker 5 hours to travel 10 miles upstream and 2 hours to travel 10 miles downstream. The speed of the current remains constant during the trip. Find the average speed of the kayak in still water and the speed of the current. ANSWER average speed of the kayak: 3.5 mi/h, speed of the current 1.5 mi/h Daily Homework Quiz Warm-Up Exercises Solve the linear system using elimination. 1. –5x + y = 18 3x – y = –10 ANSWER 2. 4x + 2y = 14 4x – 3y = –11 ANSWER 3. (–4, –2) (1, 5) 2x – y = –14 y = 3x + 6 ANSWER (8, 30) Daily Homework Quiz Warm-Up Exercises 4. x + 4y = 15 2y = x – 9 ANSWER (11, 1) 5. A business center charges a flat fee to send faxes plus a fee per page. You send one fax with 4 pages for $5.36 and another fax with 7 pages for $7.88. Find the flat fee and the cost per page to send a fax. ANSWER flat fee: $2, price per page: $.84