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

Warm-Up 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 3 h

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

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

Substitute 1 for x and 3 for y in each of the original equations. Example 1 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

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

Substitute 4 for x in either equation and solve for y. 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 for y. ANSWER The solution is (–4, 6).

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

Substitute 2 for x in either equation and solve for y. Example 3 STEP 4 Substitute 2 for x 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. ANSWER The solution is (2, 5).

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

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.

Example 4 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

Example 4 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 = – Equation 2: Going downstream x y 6 = +

Solve the system of equations. Example 4 STEP 2 Solve the system of equations. x – y = 4 Write Equation 1. x + y = 6 Write Equation 2. 2x = 10 Add equations. x = 5 Solve for x. Substitute 5 for x in Equation 2 and solve for y. 5 + y = 6 Substitute 5 for x in 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.

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. ANSWER average speed of the kayak: 3.5 mi/h, speed of the current 1.5 mi/h

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

Lesson Quiz 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. 5. ANSWER flat fee: $2, price per page: $0.84