Problems Involving Linear Systems Section 3.6 Problems Involving Linear Systems

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

Ex: Indicate how many solutions are in each system Slopes are different  One Intersection Slopes are the same  No Intersections Y-intercepts are different Slopes are the same Y-intercepts are the same  Infinite Solutions

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 All corresponding coefficients are in ratio and 4 times bigger 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” Only coefficients A & B are in ratio and not “C” Same Slopes but Different Y-intercepts 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: All coefficients A,B,C are in ratio Coefficients A & B - NOT in ratio Infinite Solutions One Solutions Coefficients A & B are NOT in ratio A & B are in ratio, but not “C” One Solutions 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: 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 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 1st: Indicate the Variables Let “y” be the Hourly Cost 2nd: Make the Equations Solve by Elimination The Hourly cost is $30 per hour The Annual cost is $50 per year

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? 1st: Indicate the Variables Let “x” be the Number of Adults 2nd: Make the Equations Let “y” be the Number of Students Quantity Revenue 3rd: Solve by Elimination 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? Total Investment $9000 Indicate the Variables Let “A” be amount invested in Bank A Amount $A Bank A (3%) Amount $B Bank B (5%) Let “B” be amount invested in Bank B James invested $5500 with Bank A and $3500 with Bank B Make the Equations Solve by Elimination

Indicate the Variables 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? Total Investment $10000 Indicate the Variables Let “A” be amount invested in Bank A Amount $A Bank A (7%) Amount $B Bank B (13%) Let “B” be amount invested in Bank B Make the Equations These are the interests he earned from each bank Solve by Substitution James invested $3500 with Bank A and $6500 with Bank B

Challenge: A musical charges $4. 00 for adults and $2. 50 for children 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? Adults $4.00 Kids $2.50 Total Attendance 3x 1st Night 5x 8x 2nd Night 2y 3y 5y Revenue 4 x (3x+2y) 2.5 x(5x+3y)

Challenge: A musical charges $4. 00 for adults and $2. 50 for children 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

Challenge: A musical charges $4. 00 for adults and $2. 50 for children 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? Adults $4.00 Kids $2.50 Total Attendance 3(80) 5(80) 1st Night =240 =400 2(150) 3(150) 2nd Night =300 =450 Total Attended =540 adults =850 kids