6 2 Solving Systems by Substitution SOLVING SYSTEMS

6 -2 Solving Systems by Substitution SOLVING SYSTEMS OF EQUATIONS ALGEBRAICALLY Word Problems Holt Algebra 11

6 -2 Solving Systems by Substitution Example 1: Consumer Economics Application Jenna is deciding between two cell-phone plans. The first plan has a $50 sign-up fee and costs $20 per month. The second plan has a $30 sign-up fee and costs $25 per month. After how many months will the total costs be the same? What will the costs be? If Jenna has to sign a one-year contract, which plan will be cheaper? Explain. Write an equation for each option. Let y represent the total amount paid and x represent the number of months. Holt Algebra 1

6 -2 Solving Systems by Substitution Example 1 Continued Total paid Payment for each is amount month. plus Sign up fee Option 1 y = $20 x + $50 Option 2 y = $25 x + $30 Step 1 y = 20 x+50 y = 25 x + 30 Both equations are solved for y (y=mx + b) Step 2 20 x + 50 = 25 x + 30 Set the 2 equations equal to each other which eliminates the y Holt Algebra 1

6 -2 Solving Systems by Substitution Example 1 Continued Step 3 20 x + 50 = 25 x + 30 – 25 x -5 x + 50 = 30 – 50 -5 x = -20 -5 -5 x=4 Solve for x. Subtract 25 x from both sides. Subtract 50 from both sides. Step 4 y = 25 x + 30 Write one of the original equations. Substitute 4 for x. Simplify. y = 25(4) + 30 y = 100 + 30 y = 130 Holt Algebra 1 Divide both sides by -5.

6 -2 Solving Systems by Substitution Example 1 Continued Step 5 (4, 130) Write the solution as an ordered pair. In 4 months, the total cost for each option would be the same $130. If Jenna has to sign a one-year contract, which plan will be cheaper? Explain. Option 1: y = 20(12) + 50 = 290 Option 2: y = 25(12) + 30 = 330 Jenna should choose the first plan because it costs $290 for the year and the second plan costs $330. Holt Algebra 1

6 -2 Solving Systems by Substitution Example 2 One cable television provider has a $60 setup fee and $80 per month, and the second has a $160 equipment fee and $70 per month. a. In how many months will the cost be the same? What will that cost be. Write an equation for each option. Let y represent the total amount paid and x represent the number of months. Holt Algebra 1

6 -2 Solving Systems by Substitution Example 2 Continued Total paid payment is amount Option 1 y = Option 2 y = $80 $70 Step 1 y = 80 x + 60 y = 70 x + 160 for each plus month. fee x + $60 x + $160 Both equations are solved for y (y=mx + b. Step 2 80 x + 60 = 70 x + 160 Set both equations equal to each other to eliminate the y’s. Holt Algebra 1

6 -2 Solving Systems by Substitution Example 2 Continued Step 3 80 x + 60 = 70 x + 160 Solve for x. Subtract 70 m – 70 x from both sides. 10 x + 60 = 160 Subtract 60 from both – 60 sides. 10 x = 100 Divide both sides by 10. 10 10 x = 10 Write one of the original Step 4 y = 70 x + 160 equations. y = 70(10) + 160 Substitute 10 for m. y = 700 + 160 Simplify. y = 860 Holt Algebra 1

6 -2 Solving Systems by Substitution Example 2 Continued Step 5 (10, 860) Write the solution as an ordered pair. In 10 months, the total cost for each option would be the same, $860. b. If you plan to move in 6 months, which is the cheaper option? Explain. Option 1: y = 60 + 80(6) = 540 Option 2: y = 160 + 270(6) = 580 The first option is cheaper for the first six months. Holt Algebra 1

6 -2 Solving Systems by Substitution Example 3 3. Plumber A charges $60 an hour. Plumber B charges $40 to visit your home plus $55 for each hour. For how many hours will the total cost for each plumber be the same? How much will that cost be? If a customer thinks they will need a plumber for 5 hours, which plumber should the customer hire? Explain. 8 hours; $480; plumber A: plumber A is cheaper for less than 8 hours. Holt Algebra 1

6 -2 Solving Systems by Substitution Example 5 4. The Strauss family is deciding between two lawn -care services. Green Lawn charges a $49 start -up fee, plus $29 per month. Grass Team charges a $25 start-up fee, plus $37 per month. a. In how many months will both lawn-care services cost the same? b. If the family will use the service for only 6 months, which is the better option? Explain. 3, $136; Green Lawn: for 6 months, Green Lawn’s service cost only $223 while Green Team’s cost $247 Holt Algebra 1

6 -2 Solving Systems by Substitution 5. ) Jack and Jason are saving for new scooters. So far, Jack has saved $9 and can earn $6 per hour dogsitting. Jason has saved $3 and can earn $9 per hour working as a lifeguard. After how many hours of work will Jack and Jason have saved the same amount? What will that amount be? Solve by substitution showing all work and your check. (2, 21) Holt Algebra 1

6 -2 Solving Systems by Substitution 6. ) Angus runs 7 miles per week and increases his distance by 1 mile each week. Myles runs 4 miles per week and increases his distance by 2 miles each week. In how many weeks will Angus and Myles be running the same distance? What will that distance be? (3, 10) Holt Algebra 1

6 -2 Solving Systems by Substitution 8. ) Derek is going to have a party catered. Good Eats charges $120 plus $10 person. Food Fare charges $150 plus $8 person. Find the # of people for which the total cost is the same for both catering companies. (15, 170) Holt Algebra 1

6 -2 Solving Systems by Substitution More difficult word problems Holt Algebra 1

6 -2 Solving Systems by Substitution Example 8 A jar contains x nickels and y dimes. There are 20 coins in the jar, and the total value of the coins is $1. 40. How many nickels and how many dimes are in the jar? (Hint: nickels are worth $0. 05 and dimes are worth $0. 10 Write an equation for number of coins in the jar. Let x represent the total number of nickels and y represent the number of dimes Write an equation for the total money in the jar. Let x represent the total number of nickels and y represent the number of dimes Holt Algebra 1

6 -2 Solving Systems by Substitution Example 8 Continued nickels Option 1 Option 2 Step 1 Holt Algebra 1 plus dimes x + $0. 05 x + y $0. 10 y is = = Total in jar 20 $1. 40 Solve 1 st equation for “y” ; x + y= 20 Slope intercept form (y=mx + b) -x -x y= -x+20

6 -2 Solving Systems by Substitution Example 8 Continued Step 1 0. 05 x +. 10 y= 1. 40 Solve 2 nd equation for “y” ; -0. 05 x Slope intercept form (y=mx + b). 10 y= -. 05 x+1. 40. 10. 10 Step 2 Holt Algebra 1 Set the 2 equations equal to each other eliminating the y’s.

6 -2 Solving Systems by Substitution Example 8 Continued Step 3 -x + 20 = -½x + 14 Solve for x. Add ½ to +½x both sides. -. 50 x + 20 = 14 Subtract 20 from both -20 – 20 sides. -. 50 x = -6. 00 Divide both sides by -. 50 -. 5 x = 12 Write one of the original Step 4 y = -x + 20 equations. y = -(12)+ 20 Substitute 12 for x. y = -12 + 20 Simplify. y=8 (12, 8) 12 nickels, 8 dimes Holt Algebra 1

6 -2 Solving Systems by Substitution Example 9 A landscaping company placed two orders with a nursery. The first order was for 13 bushed and 4 trees and totaled $487. The second order was for 6 bushes and 2 trees, and totaled $232. The bills do not list the cost per bush or tree. What were the cost of one bush and one tree? Write an equation for the total cost for the first order. Let x represent the total number of bushes and y represent the total number of trees Write an equation for the total cost for the second order. Let x represent the total number of bushes and y represent the total number of trees Holt Algebra 1

6 -2 Solving Systems by Substitution Example 9 Continued bushes Order 1 Order 2 Step 1 Holt Algebra 1 plus 13 x + 6 x + trees is Total of orders 4 y = $487 = $232 2 y Solve 1 st equation 13 x + 4 y= 487 for “y” ; -13 x Slope intercept 4 y= -13 x + 487 form (y=mx + b) 4 4 4 y= -$3. 25 + 121. 75

6 -2 Solving Systems by Substitution Example 9 Continued Solve 2 nd equation Step 1 6 x + 2 y= 232 for “y” ; -6 x Slope intercept form 2 y= -6 x+ 232 (y=mx + b) 2 2 2 Step 2 Holt Algebra 1 Set the 2 equations equal to each other eliminating the y’s.

6 -2 Solving Systems by Substitution Example 9 Continued Step 3 -3. 25 x + 121. 75 = -3 x + 116 +3. 00 x +3 x -. 25 x + 121. 75 = 116 -121. 75 -. 25 x = -5. 75 -. 25 x = 23 Step 4 6 x + 2 y = 232 Solve for x. Add 3 x to both sides. Subtract 20 from both sides. Divide both sides by -. 25 1 bush cost $23. 00 Write one of the original equations. Substitute 23 for x. 6(23) + 2 y = 232 Solve for y 138 + 2 y = 232 (23, 47) 1 bush cost $23; 1 y= 47 tree cost $47 Holt Algebra 1
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