Name Class Date 5 Solving Special Sstems Going Deeper Essential question: How do ou solve sstems with no or infinitel man solutions? 1 A-REI.3.6 EXAMPLE Solving Special Sstems b Graphing Use the graph to solve each sstem of linear equations. 3 - = 1 A { + = 7 + = 6 Is there a point of intersection? Eplain. + = 3 + = 6 6 + = 7 Does this linear sstem have a solution? Use the graph to eplain. - 0-6 B { + = 6 + = 3 Is there a point of intersection? Eplain. Does this linear sstem have a solution? Use the graph to eplain. REFLECT 1a. Use the graph to identif two lines that represent a linear sstem with eactl one solution. What are the equations of the lines? Eplain our reasoning. 1b. If each equation in a sstem of two linear equations is represented b a different line when graphed, what is the greatest number of solutions the sstem can have? Eplain our reasoning. Chapter 5 91 Lesson
1c. Identif the three possible numbers of solutions for a sstem of linear equations. Eplain when each tpe of solution occurs. A-REI.3.6 EXAMPLE Solving Special Sstems Algebraicall A Solve the sstem of linear equations b substitution. { - = - - + = Step 1 Solve - = - for : = Step Substitute the resulting epression into the other equation and solve. -( ) + = Substitute the epression for the variable. 8 = Simplif. Step 3 Interpret the solution. Graph the equations to provide more information. What does the graph tell ou about the solution? 6-0 - How is this solution represented algebraicall when the sstem is solved using substitution? B Solve the sstem of linear equations b elimination. { + = - + = Step 1 Multipl the first equation b -. -( + = -) + (-) = Step Add the new equation from Step 1 to the original second equation. + (-) = + + = 0 + 0 = 0 0 = 0 Chapter 5 9 Lesson
Step 3 Interpret the solution. Graph the equations to provide more information. What does the graph tell ou about the solution? -6-0 - How is this solution represented algebraicall when the linear sstem is solved using substitution? -6 REFLECT a. If represents a variable and a and b represent constants such that a b, interpret what each result means when solving a sstem of linear equations b substitution. = a a = b a = a b. In part B of Eample, wh is it more efficient to solve and substitute for than to solve and substitute for? c. Give two possible solutions of the sstem in part B of Eample. How are all the solutions of this sstem related to one another? PRACTICE Solve each sstem b graphing. Check our answer. 1. { + = - - = -6-0 - -6. { - = -6 - = -1 8 0 8 Chapter 5 93 Lesson
Solve each sstem b substitution. Check our answer. 3. { - = 5 - = 9. { - = = + 8 Tell whether it is more efficient to solve for and then substitute for or to solve for and then substitute for. Eplain our reasoning. Then solve the sstem. 5. { _ + = 6 + = 3 Solve each sstem b adding or subtracting. Check our answer. 6. { + = -3 - = - 7. { - 6 = 7 - + 6 = -7 8. If a linear sstem has no solution, what happens when ou tr to solve the sstem b adding or subtracting? 9. If a linear sstem has infinitel man solutions, what happens when ou tr to solve the sstem b adding or subtracting? Solve each sstem b multipling. Check our answer. 10. { + 3 = -6 10 + 15 = -30 11. { 3 - = -1-6 + 8 = 3 Chapter 5 9 Lesson
Name Class Date Additional Practice 5 = 3 1. = 3. 3 + = 3 = 7 3. = + 1 = 6. + 3 = 0 = + 3 = 3( 1) = 5 5. 6. + 3 = 3 = 3 7. Sabina and Lou are reading the same book. Sabina reads 1 pages a da. She had read 36 pages when Lou started the book, and Lou reads at a pace of 15 pages per da. If their reading rates continue, will Sabina and Lou ever be reading the same page on the same da? Eplain. 8. Brandon started jogging at miles per hour. After he jogged 1 mile, his friend Anton started jogging along the same path at a pace of miles per hour. If the continue to jog at the same rate, will Anton ever catch up with Brandon? Eplain. Chapter 5 95 Lesson
Problem Solving Write the correct answer. 1. Tra and Charmian are training for a bike race. Tra has logged 56 miles so far and rides 8 miles per week. Charmian has logged 15 miles so far and rides 8 miles per week. If these rates continue, will Tra s distance ever equal Charmian s distance? Eplain. 3. The Singhs start savings accounts for their twin bos. The accounts earn 5% annual interest. The initial deposit in each account is $00. Classif this sstem and find its solution, if an.. Metroplepress and Local Epress are courier companies. Metroplepress charges $15 to pick up a package and $0.50 per mile. Local Epress charges $10 to pick up a package and $0.55 per mile. Classif this sstem and find its solution, if an.. Frank earns $8 per hour. Madison earns $7.50 per hour. Frank started working after Madison had alread earned $300. If these rates continue, will Frank s earnings ever equal Madison s earnings? If so, when? Select the best answer. 5. A studio apartment at The Oaks costs $00 per month plus a $350 deposit. A studio apartment at Crossroads costs $00 per month plus a $300 deposit. How man solutions does this sstem have? A no solutions B 1 solution C solutions D an infinite number of solutions 7. A tank filled with 75 liters of water loses 0.5 liter of water per hour. A tank filled with 50 liters of water loses 0.1 liter of water per hour. How would this sstem be classified? A inconsistent B dependent C consistent and independent D consistent and dependent 6. Jane and Gar are both landscape designers. Jane charges $75 for a consultation plus $5 per hour. Gar charges $50 for a consultation plus $30 per hour. For how man hours will Jane s charges equal Gar s charges? F never G after hours H after 5 hours J alwas 8. Simon is 3 ears older than Renata. Five ears ago, Renata was half as old as Simon is now. How old are Simon and Renata now? F Simon is 13 and Renata is 10. G Simon is 15 and Renata is 10. H Simon is 16 and Renata is 8. J Simon is 16 and Renata is 13. Chapter 5 96 Lesson