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Warm Up

Solve Linear Systems by Adding or Subtracting. Warm Up. Lesson Presentation. Lesson Quiz. (5, 2). ANSWER. 3 h. ANSWER. Warm-Up. 1. Solve the linear system using substitution. 2 x + y = 12 3 x – 2 y = 11.

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Warm Up

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  1. Solve Linear Systems by Adding or Subtracting Warm Up Lesson Presentation Lesson Quiz

  2. (5, 2) ANSWER 3 h ANSWER Warm-Up 1. Solve the linear system using substitution. 2x + y = 12 3x – 2y = 11 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?

  3. Add the equations to 2x + 3y = 11 eliminate one variable. –2x + 5y = 13 Example 1 Solve the linear system: 2x + 3y = 11 Equation 1 –2x + 5y = 13 Equation 2 SOLUTION STEP 1 STEP 2 Solve fory. 8y =24 y =3

  4. ANSWER The solution is(1, 3). Example 1 STEP 3 Substitute 3 for yin either equation and solve for x. 2x + 3y = 11 Write Equation 1 2x + 3(3)= 11 Substitute3for y. x = 1 Solve for x.

  5. ? ? 2(1)+ 3(3)= 11 2(1)+ 5(3)= 13 Example 1 Substitute 1 for xand 3 foryin each of the original equations. CHECK 2x + 3y= 11 2x + 5y= 13 11= 11 13= 13

  6. Subtract the equations to 4x + 3y = 2 eliminate one variable. 5x + 3y = –2 Example 2 Solve the linear system: 4x + 3y = 2 Equation 1 5x + 3y = –2 Equation 2 SOLUTION STEP 1 STEP 2 Solve forx. – x = 4 x = 4

  7. ANSWER The solution is (–4, 6). Example 2 STEP 3 Substitute4 for x in either equation and solve for y. 4x+ 3y = 2 Write Equation 1. 4(–4)+ 3y = 2 Substitute –4 for x. y = 6 Solve fory.

  8. 3x + 4y = 14 Add the equations. 5x = 10 Solve for x. x = 2 Example 3 Solve the linear system: 8x – 4y = –4 Equation 1 4y = 3x + 14 Equation 2 SOLUTION Rewrite Equation 2 so that the like terms are arranged in columns. STEP 1 8x –4y = –4 8x – 4y = –4 4y = 3x + 14 STEP 2 STEP 3

  9. ANSWER The solution is (2, 5). Example 3 STEP 4 Substitute 2 forx in either equation and solve for y. 4y = 3x + 14 Write Equation 2. 4y = 3(2)+ 14 Substitute 2 for x. y = 5 Solve for y.

  10. – 5x – 6y = 8 – 5y 2y 3x + 4y = 1 4x + 6 3x + 6 = = ANSWER (2, –3) ANSWER (1, 2) (5, 4) ANSWER ANSWER (1, 1) ANSWER (–2, 0) ANSWER (–1, –3) Guided Practice Solve the linear system. 1. 4x – 3y = 5 2. –2x + 3y = –7 5x + 2y = 4 3. 4. 6x – 4y = 14 7x – 2y = 5 7x – 3y = 4 5. 6. 3x + 4y = –6 2x + 5y = 12

  11. Example 4 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.

  12. Upstream: Downstream: 12 =r 3 12 =r 2 d = rt d =rt Example 4 STEP1 Write a system of equations. First find the speed of the kayak going upstream and the speed of the kayak going downstream. 4 =r 6 =r

  13. Equation 2: Going downstream Going upstream Equation 1: x – y 6 x y = = + 4 Example 4 Use the speeds to write a linear system. Let xbe the average speed of the kayak in still water, and let ybe the speed of the current.

  14. x + y = 6 Example 4 STEP2 Solve the system of equations. x –y = 4 Write Equation 1. Write Equation 2. 2x = 10 Add equations. x = 5 Solve forx. Substitute 5 for xin Equation 2 and solve for y. 5 + y = 6 Substitute 5 for xin Equation 2. y = 1 Subtract 5 from each side. The average speed of the kayak in still water is 5 miles per hour, and the speed of the current is 1 mile per hour.

  15. ANSWER average speed of the kayak: 3.5 mi/h, speed of the current 1.5 mi/h Guided Practice 7. 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.

  16. 4. x + 4y = 15 1. –5x + y = 18 3x–y = –10 2y = x – 9 ANSWER (–4, –2) ANSWER (11, 1) 2. 4x + 2y = 14 4x – 3y = –11 ANSWER (1, 5) 3. 2x – y = –14 y = 3x + 6 ANSWER (8, 30) Lesson Quiz Solve the linear system using elimination.

  17. 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: $0.84 Lesson Quiz

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