Linear Systems Problem-solving Techniques

Section 3.6 Problems Involving Linear Systems

I) Number of Solutions: y 4 1. NO SOLUTIONS (System is Inconsistent) 2 x Lines are parallel and do not intersect each other Lines have the same slope but different y-intercept -4 -3 -2 -1 0 1 2 3 4 -2 -4 y 2. ONE SOLUTION (System is Consistent) 4 Lines intersect at ONE point 2 Lines have Different slopes x -4 -3 -2 -1 0 1 2 3 4 y (System is Consistent) 3. INFINITE SOLUTIONS 4 Lines OVERLAP each other 2 Lines have both the SAME slope and Y-intersect x -4 -3 -2 -1 0 1 2 3 4

Ex: Indicate how many solutions are in each system = = + + 5 5 y y x x 3 2 1 2 One Intersection Slopes are different = = + 3 5 y y 2 x Slopes are the same Y-intercepts are different No Intersections 3 4 x 6 = = + 5 6 y y x Slopes are the same Y-intercepts are the same Infinite Solutions = + + 10 x 12 2 10 12 x 2 2

+ = Ax By C II) Number of Solutions in Standard Form: A Linear System with INFINITE Solutions will have all 3 coefficients A,B,C in ratio with the same constant 3 2 6 x y = 4 4 4 4 times bigger All corresponding coefficients are in ratio and = 12 8 24 x y Same Slopes and Same Y-intercept A Linear System with NO Solutions will only have coefficients A & B in ratio with the same constant and NOT C 3 2 6 x y = 3 4 4 Same Slopes but Different Y-intercepts Only coefficients A & B are in ratio and not C = 12 8 18 x y A Linear System with ONE Solution will have different ratios for coefficients A & B. C doesn t matter.

Ex: Indicate the Number of Solutions in each Linear System: = 3 5 4 x y ( + = 2 6 11 x y ( ) 2 ) 3 2 3 ( ) 2 + = + = 4 12 22 x y 9 15 12 x y Coefficients A & B - NOT in ratio One Solutions All coefficients A,B,C are in ratio Infinite Solutions = 6 3 12 x y 3 4 = 3 8 10 x y 3 ( ) 3 4 1 2 4 ? = 9 x y 9 2 9 4 + = 1.5 5 5 x y Coefficients A & B are NOT in ratio One Solutions A & B are in ratio, but not C NO Solutions

III) Solving Problems Involving Linear Systems In the next part, you will have scenarios that involves problems with linear equations First step: Indicate What the Variables are Number of People Cost for a certain item 2nd Step: Read the Question to generate your 2 equations Revenue Interest Earned: I PRT = = P Q R = Price Quantity Revenue : : : : I Interest Earned P Principle R Interest Rate Decimal Form ( # T Time of Years ( ) $ at Beginning ( ) ) 3rd Step: Solve the system by Elimination or Substitution 4th Step: Write your concluding statement

Ex: A tutoring center charges an annual fee and an hourly fee. 8 hours of tutoring cost $290. 15 hours cost $500. Find the annual cost and hourly cost. Let x be the Annual Cost Let y be the Hourly Cost = 1st: Indicate the Variables 2nd: Make the Equations + Cost Annual Fee Hourly Fees Solve by Elimination + + = 8 $290 = = $30 y = ) 30 = x = x y 15 $500 x y $210 0 7y The Hourly cost is $30 per hour The Annual cost is $50 per year ( x+ 8 $290 $50

Practice: The cost for a school play is $35 per adult and $20 per student. 160 people attended the play and total revenue was $4100. How many students and adults attended? Let x be the Number of Adults Let y be the Number of Students 160 x y + = 35 20 4100 x y + = ( Revenue: Total 1st: Indicate the Variables 2nd: Make the Equations ( ) Total people Quantity ) = P Q R Revenue 3rd: Solve by Elimination 5600 4100 1500 = 100 y = 60 x = + + + = = + x = 35 35 35 20 15y x x y y ( ) 160 20 y x y + 3 = 35 4100 0 100 students and 60 parents attended the school play

Ex: James invested $9000, part with Bank A (3%) and part with Bank B (5%). After one year, he made a total of $340 in interest. How much did he invest with each bank? Indicate the Variables Let A be amount invested in Bank A $340 70 = 3500 B = 5500 A= Total Investment $9000 $9000 B = 0.03 0.05 A + ( ) + + = = + 0.03 0.03 0.03 0.03 0.05 0.02B $270 A A B B A = $340 B Amount $A Bank A (3%) ( )( A = 0.03 = Amount $B Bank B (5%) ( )( B = 0.05 I = 0 Let B be amount invested in Bank B James invested $5500 with Bank A and $3500 with Bank B ) ) 0.03 1 I 0.05 1 I Make the Equations A B + 0.05 A B + I A B = = $9000 $340 0.03 Solve by Elimination

Ex: James invested $10000, part with Bank A (7%) and part with Bank B (13%). After one year, both banks made the same amount of Interest. How much did he invest with each bank? $10000 A B + = 0.07 0.13 A B = ( ) 10000 0.07 0.13 B B = 700 0.07 0.13 B B = 700 0.20B = 3500 B = 6500 A = These are the interests he earned from each bank = $10000 A B Indicate the Variables Let A be amount invested in Bank A Total Investment $10000 Amount $A Bank A (7%) ( )( A = 0.07 = Amount $B Bank B (13%) ( )( B = 0.13 I = Let B be amount invested in Bank B ) ) 0.07 1 I 0.13 1 B I Make the Equations A B + = 0.07 A = Solve by Substitution I A $10000 0.13 B James invested $3500 with Bank A and $6500 with Bank B

Challenge: A musical charges $4.00 for adults and $2.50 for children. On the first night, the ratio of adults to kids was 3:5. On the second night, the ratio was 2:3. A total of 1390 people attended for two nights, and the revenue generated was $4285. How many adults and kids attended in each night? Total Kids $2.50 Adults $4.00 Attendance 8x 3x 5x 1st Night 2y 2nd Night 5y 3y 2.5 x(5x+3y) Revenue 4 x (3x+2y)

Challenge: A musical charges $4.00 for adults and $2.50 for children. On the first night, the ratio of adults to kids was 3:5. On the second night, the ratio was 2:3. A total of 1390 people attended for two nights, and the revenue generated was $4285. How many adults and kids attended? Make the Equations Quantity Revenue + + = ) 8 ( 5 2 1390 2.50 5 + 24.50 = 15.50 x y y ( ) y + = = ) 4 3 3 4285 4285 x x 15.50 ( y + x + + = = + + 24.80 24.50 15.50 15.50 0.30 4309 4285 24 80 x x y y x = x = 8 5 1390 y x x y 3.1 = 24.50 4285 y = 150

Challenge: A musical charges $4.00 for adults and $2.50 for children. On the first night, the ratio of adults to kids was 3:5. On the second night, the ratio was 2:3. A total of 1390 people attended for two nights, and the revenue generated was $4285. How many adults and kids attended? Total Kids $2.50 5(80) =400 Adults $4.00 3(80) =240 Attendance 1st Night 2(150) =300 3(150) =450 2nd Night Total Attended =850 kids =540 adults