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1. Solve the linear system using substitution. 2 x + y = 12 3 x – 2 y = 11

Learn how to solve linear systems using substitution and elimination methods. Calculate auto repair charges with varying labor costs at different service shops. Practice solving guided examples to improve your skills.

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1. Solve the linear system using substitution. 2 x + y = 12 3 x – 2 y = 11

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  1. (5, 2) ANSWER 3 h ANSWER 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?

  2. Add the equations to 2x + 3y = 11 eliminate one variable. –2x + 5y = 13 EXAMPLE 1 Use addition to eliminate a variable 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

  3. ANSWER The solution is(1, 3). EXAMPLE 1 Use addition to eliminate a variable 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.

  4. ? ? 2(1)+ 3(3)= 11 −2(1)+ 5(3)= 13 EXAMPLE 1 Use addition to eliminate a variable CHECK Substitute 1 for xand 3 foryin each of the original equations. 2x + 3y= 11 −2x + 5y= 13 11= 11 13= 13

  5. Subtract the equations to 4x + 3y = 2 eliminate one variable. 5x + 3y = –2 EXAMPLE 2 Use subtraction to eliminate a variable 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

  6. ANSWER The solution is (–4, 6). EXAMPLE 2 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. 4(–4)+ 3y = 2 Substitute –4 for x. y = 6 Solve fory.

  7. −3x + 4y = 14 Add the equations. 5x = 10 Solve for x. x = 2 EXAMPLE 3 Arrange like terms 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

  8. ANSWER The solution is (2, 5). EXAMPLE 3 Arrange like terms 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.

  9. ANSWER (–1, –3) for Example 1,2 and 3 GUIDED PRACTICE Solve the linear system: 1. 4x – 3y = 5 ` –2x + 3y = –7

  10. – 5x – 6y = 8 ANSWER (2, –3) for Example 1,2 and 3 GUIDED PRACTICE Solve the linear system: 2. 5x + 2y = 4

  11. – 3x + 4y = 1 (5, 4) ANSWER for Example 1,2 and 3 GUIDED PRACTICE Solve the linear system: 3. 6x – 4y = 14

  12. ANSWER (1, 1) for Example 1,2 and 3 GUIDED PRACTICE Solve the linear system: 4. 7x – 2y = 5 7x – 3y = 4

  13. 2y 3x + 6 = ANSWER (–2, 0) for Example 1,2 and 3 GUIDED PRACTICE Solve the linear system: 5. 3x + 4y = –6

  14. 5y 4x + 6 = ANSWER (1, 2) for Example 1,2 and 3 GUIDED PRACTICE Solve the linear system: 6. 2x + 5y = 12

  15. 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. EXAMPLE 4 Write and solve a linear system KAYAKING

  16. Upstream: Downstream: 12 =r 3 12 =r 2 d = rt d =rt EXAMPLE 4 Write and solve a linear system 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

  17. Going upstream Equation 1: – y x = 4 EXAMPLE 4 Write and solve a linear system 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.

  18. Equation 2: Going downstream x 6 y = + EXAMPLE 4 Write and solve a linear system

  19. x + y = 6 EXAMPLE 4 Write and solve a linear system 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.

  20. EXAMPLE 4 Write and solve a linear system 5 + y = 6 Substitute 5 for xin Equation 2. y = 1 Subtract 5 from each side. ANSWER 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.

  21. ANSWER average speed of the kayak: 3.5 mi/h, speed of the current: 1.5 mi/h for Example 4 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.

  22. 1. –5x + y = 18 3x–y = –10 ANSWER (–4, –2) 2. 4x + 2y = 14 4x – 3y = –11 ANSWER (1, 5) 3. 2x – y = –14 y = 3x + 6 ANSWER (8, 30) Daily Homework Quiz Solve the linear system using elimination.

  23. 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 Daily Homework Quiz

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