Modelling and Solving Two-Step Equation: a( + b) c Focu on After thi leon, you will be able to model problem with two-tep linear equation olve two-tep linear equation and how how you worked out the anwer Kia plan to make a quare Star Quilt for her grandmother. The quilt will have a 4-cm wide border around it. Kia want the perimeter of the completed quilt to be 600 cm. How can Kia decide how long each ide of the quilt hould be before he add the border? How do you olve equation of the form a( + b) c? algebra tile Viktor mied yeterday math cla. Jackie will how him how to model and olve the equation 3( - 5) -6 1. a) Ue a variable tile to repreent. b) How will you ue negative 1-tile to repreent -5? 2. a) How many et of - 5 will you include in your model? Eplain. b) How will you complete your model of the equation? The centre of a Star Quilt i in the hape of the traditional eight-pointed morning tar of the Lakota and Dakota Siou. The Star Quilt i a ymbol of tradition to the Plain people. 3. a) What i the firt thing you do to iolate the variable tile? b) What equation doe your model repreent now? c) What do you need to do to olve the equation? 4. What i the unknown value of? Reflect on Your Finding 5. What tep did you take to olve the equation? 394 MHR Chapter 10
Eample 1: Model With Algebra Tile A flower garden i in the hape of a rectangle. The length of the garden i 2 m longer than the length of the hed beide it. The width of the garden i 4 m. If the area of the garden i 20 m2, what i the 4 m length of the hed? 2m A 20 m2 Solution Let repreent the unknown length of the hed. The length of the garden can be repreented by + 2. The width of the garden i 4 m. The equation that model the area of the garden i 4( + 2) 20. There are four group of ( + 2). That mean there are four variable tile and eight poitive 1-tile on the left ide of the equation. To iolate the variable, ubtract eight poitive 1-tile from both ide of the equal ign. Thi i the ame a adding eight negative 1-tile to both ide of the equal ign. There are now four variable tile on the left ide and 12 poitive 1-tile on the right ide. The four variable tile mut have the ame value a the 12 poitive 1-tile. Each variable tile mut then have a value of three poitive 1-tile. The length of the hed i 3 m. Check: Left Side 4( + 2) Right Side 20 4(3 + 2) 4(5) 20 Left Side Right Side The olution i correct. Solve by modelling the equation. a) 2(g + 4) -8 b) 3(r - 2) 3 What i the formula for the area of a rectangle? Literacy Link When ubtituting a value into an equation, be ure to ue the correct order of operation: Bracket. Multiply and divide in order from left to right. Add and ubtract in order from left to right. 10.4 Modelling and Solving Two-Step Equation: a(+ b) c MHR 10_ML8_Chapter10_10th_B.indd 395 395 4/9/08 4:03:43 PM
Eample 2: Solve Equation Kia i making a quare quilt with a 4-cm wide border around it. She want the completed quilt to have a perimeter of 600 cm. What mut the dimenion of Kia quilt be before he add the border? Strategie Draw a Diagram The length mut be multiplied by 4 becaue there are four ide to the quare quilt. Literacy Link The ditributive property tate that a(b + c) equal a b + a c. To ue the ditributive property, multiply the term in the bracket by 4. 4( + 8) 4 ( + 8) (4 ) + (4 8) 4 + 32 Solution Let repreent the unknown ide length of the quilt before the border i added. A border of 4 cm i added to each ide. That mean the ide length of the quilt after the border i added i + 8. Model with the equation 4( + 8) 600. Method 1: Divide Firt Iolate the variable. 4( + 8) 600 4( + 8) 4 600 4 + 8 Divide by 4 to undo the multiplication. + 8 150 + 8-8 150-8 Subtract 8 to undo the addition. 142 The quilt dimenion before adding the border hould be 142 cm 142 cm. Method 2: Ue the Ditributive Property Firt Iolate the variable. 4( + 8) 600 4 + 32 600 Multiply both and 8 by 4. 4 + 32-32 600-32 Subtract 32 from both ide of the equation. 4 568 4 4 568 Divide both ide of the equation by 4. 4 142 The quilt dimenion before adding the border hould be 142 cm 142 cm. Check: Left Side 4( + 8) Right Side 600 4(142 + 8) 4(150) 600 Left Side Right Side The olution i correct. 4 cm 4 cm Solve each equation. a) -2( - 3) 12 b) -20 5(3 + p) 396 MHR Chapter 10
To olve an equation, iolate the variable on one ide of the equal ign. When undoing the operation performed on the variable, ue oppoite operation. Solve an equation of the form a( + b) c by dividing firt or by uing the ditributive property firt. Divide Firt: -4( - 7) 16-4( - 7) -4 16-4 Divide by -4 to undo the multiplication. - 7-4 - 7 + 7-4 + 7 Add 7 to undo the ubtraction. 3 Ue the Ditributive Property Firt: -4( - 7) 16-4 + 28 16 Ue the ditributive property. -4 + 28-28 16-28 Subtract 28 to undo the addition. -4-12 -4-4 -12-4 Divide by -4 to undo the multiplication. 3 One method you can ue to check your anwer i ubtituting it back into the equation. Both ide of the equation hould have the ame value. Left Side -4( - 7) Right Side 16-4(3-7) -4(-4) 16 Left Side Right Side The olution i correct. 1. Draw diagram to how how you would olve the equation 4 2(v - 3) uing algebra tile. Eplain each tep in word. 2. Julia and Chri are olving the equation -18-6( + 2). I either trategy correct? Eplain. Julia: Firt, I ubtract 2 from both ide. Then, I divide both ide by 6. Chri: I tart by dividing 6( + 2) by 6. Then, I ubtract 2 from both ide. 3. Decribe a ituation that can be modelled with the equation 2(r + 3) -6. 10.4 Modelling and Solving Two-Step Equation: a(+ b) c MHR 397
For help with #4 to #7, refer to Eample 1 on page 395. 4. Solve the equation modelled by each diagram. Check your olution. a) b) 5. Solve the equation repreented by each diagram. Verify your anwer. a) b) 6. Model and then olve each equation. Check your anwer. a) 3(t - 2) 12 b) 6(j - 1) -6 7. Model and then olve each equation. Verify your olution. a) 2(3 + p) 8 b) 0 7(n - 2) For help with #8 and #9, refer to Eample 2 on page 396. 8. Solve each equation. Check your anwer. a) 6(r + 6) -18 b) 4(m - 3) 12 c) 3(1 + g) -75 d) 36 6(f + 13) 9. Solve. Verify your olution. a) -21 3(k + 3) b) 42-14(n - 11) c) 8( - 7) -32 d) -10-5(w + 13) 10. Show whether -4 i the olution to each equation. a) -8( - 1) 24 b) 3(-8 - ) -24 c) 25-5( - 1) d) 66 6( + 7) 11. The fence around Giel new tree i in the hape of an equilateral triangle. Giel want to increae the length of each ide by 7 cm. The perimeter of her new fence will be 183 cm. a) Let repreent the ide length of the old fence. What equation model thi ituation? b) Determine the length of each ide of the old fence. 12. The amount of food energy per day required by hiker i modelled by the equation e -125(t - 122), where e i the amount of food energy, in kilojoule (kj), and t i the outide temperature, in degree Celiu. a) If the outide temperature i -20 C, how much food energy i required per day? b) If a hiker conume 19 000 kj of food energy baed on the outide temperature, what i the temperature? 398 MHR Chapter 10
13. 14. Barney want to frame a quare picture of hi dog. The frame he bought fit a picture with a perimeter no greater than 96 cm. He plan to put a 2-cm blue border around the picture. a) What equation model thi ituation? b) Determine the maimum dimenion that the picture can have. 15. A parking lot charge by the hour: $2 for the firt hour and $3 for every hour after that. The formula ued to calculate the number of hour omeone ha parked i 3(h - 1) T - 2, where h repreent a number of hour greater than zero and T repreent the total amount of the parking fee, in dollar. If Mark parking fee i $8, how long did he park in the lot? 16. The ditance between Andrew houe and hi grandfather apartment i 42 km. a) If Andrew ride hi bike 2 km/h fater than hi current peed, he could get there in 3 h. What i Andrew current peed? b) If Andrew want to get there in 2 h, how much fater than hi current peed hould he ride hi bike? c) Do you think Andrew can get there in 2 h? Eplain. A computer rental company charge by the hour: $5 for the firt hour and $4 for every hour after that. The fee rate can be modelled with the equation 4(n - 1) T - 5, where n i a number of hour greater than zero and T i the rental fee, in dollar. Candy rental fee wa $17. For how many hour did he rent the computer? MATH LINK Some job require working the night hift, uch a from midnight to 8:30 a.m. Other job require working in iolated area or under hazardou condition. Depending on the job, the wage may be increaed by a certain amount per hour or per month. Thi increae i called a premium. a) Reearch and decribe three different job that pay hourly or monthly wage plu a premium. b) For each job, model the pay uing an equation. 10.4 Modelling and Solving Two-Step Equation: a(+ b) c MHR 10_ML8_Chapter10_10th_B.indd 399 399 4/9/08 4:03:46 PM