Notes for Algebra 1 Chapter 6

System of Equations A system that contains 2 or more equations, the ordered pair that is its solution, is the solution to all equations in the system. A system can have 1 solution, many solutions, or no solutions.

Consistent/Inconsistent A system that has at least one solution./ the system has no solutions

Independent/dependent A consistent system has exactly one solution/ or infinite number of solutions.

Example 1 pg. 336 Number or solutions Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent. 1.) 𝑦=−𝑥+1 𝑦=−𝑥+4 2.) 𝑦=𝑥−3 𝑦=−𝑥+1 6 5 4 3 2 1 -3 -2 -1 𝑦=−𝑥+4 𝑦=−𝑥+1 𝑦=𝑥−3

Example 1 pg. 336 Number or solutions Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent. 1.) 𝑦=−𝑥+1 𝑦=−𝑥+4 Inconsistent 2.) 𝑦=𝑥−3 𝑦=−𝑥+1 Consistent and independent 6 5 4 3 2 1 -3 -2 -1 𝑦=−𝑥+4 𝑦=−𝑥+1 𝑦=𝑥−3

Example 2 pg. 336 Solve by graphing Graph each system and determine the number of solutions that it has. If it has one solution, name it. 1.) 𝑦=2𝑥+3 8𝑥−4𝑦=−12 2.) 𝑥−2𝑦=4 𝑥−2𝑦=−2

Example 2 pg. 336 Solve by graphing Graph each system and determine the number of solutions that it has. If it has one solution, name it. 1.) 𝑦=2𝑥+3 8𝑥−4𝑦=−12 Infinitely many solutions

Example 2 pg. 336 Solve by graphing Graph each system and determine the number of solutions that it has. If it has one solution, name it. 2.) 𝑥−2𝑦=4 𝑥−2𝑦=−2 No solutions 5 4 3 2 1 -4 -3 -2 -1

Example 3 pg. 337 Write and solve a system of equations Naresh rode 20 miles last week and plans to ride 35 miles per week. Diego rode 50 miles last week and plans to ride 25 miles per week. Predict the week in which Naresh and Diego will have ridden the same number of miles.

Example 3 pg. 337 Write and solve a system of equations 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 1 2 3 4 𝑦=25𝑥+50 Naresh rode 20 miles last week and plans to ride 35 miles per week. Diego rode 50 miles last week and plans to ride 25 miles per week. Predict the week in which Naresh and Diego will have ridden the same number of miles. Week 3 𝑦=35𝑥+20

Steps for solving systems by Substitution 1.) Solve one equation for one variable. 2.) Substitute the resulting expression from step 1 into the other equation to replace the variable and solve for the remaining variable. 3.) Then substitute the value from step 2 into either original equation. Write the answer as an ordered pair.

Example 1 pg. 344 Solve a system by substitution 1.) 𝑦=−4𝑥+12 2𝑥+𝑦=2

Example 1 pg. 344 Solve a system by substitution 1.) 𝑦=−4𝑥+12 2𝑥+𝑦=2 5, −8

Example 2 pg. 345 Solve and then Substitute 1.) 𝑥−2𝑦=−3 3𝑥+5𝑦=24

Example 2 pg. 345 Solve and then Substitute 1.) 𝑥−2𝑦=−3 3𝑥+5𝑦=24 3, 3

Example 3 pg. 345 No solution or infinitely many solutions

Example 4 pg. 346 Write and Solve a System of Equations NATURE CENTER A nature center charges $35.25 for a yearly membership and $6.25 for a single admission. Last week it sold a combined total of 50 yearly memberships and single admission for $660.50. How many memberships and how many single admission were sold?

Example 4 pg. 346 Write and Solve a System of Equations NATURE CENTER A nature center charges $35.25 for a yearly membership and $6.25 for a single admission. Last week it sold a combined total of 50 yearly memberships and single admission for $660.50. How many memberships and how many single admission were sold? 12 memberships and 38 single admission

Steps for solving systems by Elimination 1.) Write the system so the like terms with the same or opposite coefficients are aligned. 2.) Add or subtract the equations, eliminating one variable. Then solve the equation. 3.) Substitute the value from step 2 into one of the original equations and solve for the other variable. Write the solution as an ordered pair.

Example 1 pg. 350 Elimination Using Addition 1.) −3𝑥+4𝑦=12 3𝑥−6𝑦=18

Example 1 pg. 350 Elimination Using Addition 1.) −3𝑥+4𝑦=12 3𝑥−6𝑦=18 −24, −15

Example 2 pg. 351 Write and solve a system of equations Four times one number minus three times another number is 12. Two times the first number added to three times the second number is 6. Find the numbers

Example 2 pg. 351 Write and solve a system of equations Four times one number minus three times another number is 12. Two times the first number added to three times the second number is 6. Find the numbers The numbers are 3 and 0.

Example 3 pg. 352 1.) 4𝑥+2𝑦=28 4𝑥−3𝑦=18

Example 3 pg. 352 1.) 4𝑥+2𝑦=28 4𝑥−3𝑦=18 6, 2

Example 4 pg. 352 Real world examples RENTALS A hardware store earned $956.50 from renting ladders and power tools last week. The store charged customers for a total of 36 days for ladders and 85 days for power tools. This week the store charged 36 days for ladders 70 days for power tools, and earned $829. How much does the store charge per day for ladders and for power tools?

Example 4 pg. 352 Real world examples RENTALS A hardware store earned $956.50 from renting ladders and power tools last week. The store charged customers for a total of 36 days for ladders and 85 days for power tools. This week the store charged 36 days for ladders 70 days for power tools, and earned $829. How much does the store charge per day for ladders and for power tools? $6.50 per day for ladders and $8.50 per day for power tools.

Steps for solving Elimination 1.) Multiply at least one equation by a constant to get two equations that contain opposite terms. 2.) Add the equations, eliminating one variable. Then solve the equation. 3.) Substitute the value from step 2 into one of the original equations and solve for the remaining variable. Write the solution as an ordered pair.

Example 1 pg. 357 Multiply one equation to eliminate a variable 1.) 2𝑥+𝑦=23 3𝑥+2𝑦=37

Example 1 pg. 357 Multiply one equation to eliminate a variable 1.) 2𝑥+𝑦=23 3𝑥+2𝑦=37 9, 5

Example 2 pg. 358 Multiply Both Equations to Eliminate a Variable 1.) 4𝑥+3𝑦=8 3𝑥−5𝑦=−23

Example 2 pg. 358 Multiply Both Equations to Eliminate a Variable 1.) 4𝑥+3𝑦=8 3𝑥−5𝑦=−23 −1, 4

Example 3 pg. 359 solve a system of Equations TRANSPORTATION A fishing boat travels 10 miles down-stream in 30 minutes. The return trip takes the boat 40 minutes. Find the rate in miles per hour of the boat in still water. Example 2 pg. 358 Multiply Both Equations

Example 3 pg. 359 solve a system of Equations TRANSPORTATION A fishing boat travels 10 miles down-stream in 30 minutes. The return trip takes the boat 40 minutes. Find the rate in miles per hour of the boat in still water. Example 2 pg. 358 Multiply Both Equations 17.5 miles per hour

Example 1 pg. 365 choose the best method 1.) 2𝑥+3𝑦=23 4𝑥+2𝑦=34

Example 1 pg. 365 choose the best method 1.) 2𝑥+3𝑦=23 4𝑥+2𝑦=34 7, 3

Example 2 pg. 366 Apply systems of Linear equations CAR RENTAL Ace Car Rental rents a car for $45 and $0.25 per mile. Star Car Rental rents a car for $35 and $0.30 per mile. How many miles would a driver need to drive before the cost of renting the cost of renting a car at Ace Car Rental and renting a car at Star Car Rental were the same?

Example 2 pg. 366 Apply systems of Linear equations CAR RENTAL Ace Car Rental rents a car for $45 and $0.25 per mile. Star Car Rental rents a car for $35 and $0.30 per mile. How many miles would a driver need to drive before the cost of renting the cost of renting a car at Ace Car Rental and renting a car at Star Car Rental were the same? 200, 95 ; 200 miles at a cost of $95

Systems of Inequalities A set of 2 or more inequalities with the same variables.

Example 1 Solve by Graphing 1.) 𝑦<2𝑥+2 𝑦≥−𝑥−3

Example 1 Solve by Graphing 1.) 𝑦<2𝑥+2 𝑦≥−𝑥−3 5 4 3 2 1 -4 -3 -2 -1

Example 2 pg. 373 No solution 1.) 𝑦≥−3𝑥+1 𝑦≤−3𝑥−2

Example 2 pg. 373 No solution 1.) 𝑦≥−3𝑥+1 𝑦≤−3𝑥−2

Example 3 pg. 373 Who number solutions SERVICE A college service organization requires that its members maintain at least a 3.0 grade point average and volunteer at least 10 hours a week. a.)Define the variables and write a system of inequalities to represent this situation. Then graph the system. b.) Name one possible solution

Example 3 pg. 373 Who number solutions SERVICE A college service organization requires that its members maintain at least a 3.0 grade point average and volunteer at least 10 hours a week. a.)Define the variables and write a system of inequalities to represent this situation. Then graph the system. Let g = grade point average v = number of volunteer hours 𝑔≥3.0, 𝑣≥10 b.) Name one possible solution 3.5, 12 Volunteer Hours 14 12 10 8 6 4 2 1.0 2.0 3.0 4.0 Grade Point Average