SOLVING SYSTEMS OF EQUATIONS ALGEBRAICALLY Word Problems Holt Algebra 1

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.

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

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

Example 1 Continued Write the solution as an ordered pair. Step 5 (4, 130) 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.

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.

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

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

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.

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.

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

5. ) Jack and Jason are saving for new scooters 5.) Jack and Jason are saving for new scooters. So far, Jack has saved $9 and can earn $6 per hour dog-sitting. 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)

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)

(15, 170) 7.) Derek is going to have a party catered. Good Eats charges $120 plus $10 per person. Food Fare charges $150 plus $8 per person. Find the # of people for which the total cost is the same for both catering companies. (15, 170)

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

Solve 1st equation for “y” ; Slope intercept form (y=mx + b) Example 8 Continued nickels plus dimes is Total in jar Option 1 x = y + 20 Option 2 $0.05x = $0.10y + $1.40 Step 1 x + y= 20 -x -x y= -x+20 Solve 1st equation for “y” ; Slope intercept form (y=mx + b)

Example 8 Continued Step 1 0.05x + .10y= 1.40 -0.05x -0.05x .10y= -.05x+1.40 .10 .10 .10 Solve 2nd equation for “y” ; Slope intercept form (y=mx + b) Step 2 Set the 2 equations equal to each other eliminating the y’s.

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