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REVIEW: 6.1 Solving by Graphing: Remember: To graph a line we use the slope intercept form: y = mx +b Slope = − − = ⍙ = ⍙ STARING POINT (The point where it crosses the y-axis) System Solution: The point where the two lines intersect (cross): (1, 3) Remember: What are the requirements for this to happen? REVIEW: 6.2: Solving by Substitution: 0): THINK - Which variable is the easiest to isolate? 1): Isolate a variable 2): Substitute the variable into the other equation 3): Solve for the variable 4): Go back to the original equations, substitute, solve for the second variable 5): Check 6.3: Solving by Elimination: 0): THINK: Which variable is easiest to eliminate. 1): Pick a variable to eliminate 2): Add the two equations to Eliminate a variable 3): Solve for the remaining variable 4): Go back to the original equation, substitute, solve for the second variable. 5): Check NOTE: We can solve system of equations using a graph, the substitution or eliminations process. The best method to use will depend on the form of the equations and how precise we want the answer to be. CONCEPT SUMMARY: METHOD Graphing WHEN TO USE When you want a visual display of the equations, or when you want to estimate the solution. http://player.discoveryeducation.com/index.cfm?guidAs setId=8A6198F2-B782-4C69-8F6D8CD683CAF9DD&blnFromSearch=1&productcode=US YOU TRY IT: Solve the system by Graphing: −2 + = 2 6 + 2 = 14 YOU TRY IT: (SOLUTION) −2 + = 2 6 + 2 = 14 → = + → = − + (1,4) CONCEPT SUMMARY: METHOD WHEN TO USE Substitution When one equation is already solved: y=mx+b or x= ym+b . 2 x 7 2 y x2 http://www.khanacademy.org/mat h/algebra/systems-of-eq-andineq/fast-systems-ofequations/v/solving-linearsystems-bysubstitution?exid=systems_of_equ ations http://player.discoveryeducation.com/index.cfm?guidAssetId=A9199767-40AB-4AD1-9493-9391E75638D0 YOU TRY IT: Solve the system by Substitution: −2 + = 2 6 + 2 = 14 YOU TRY IT:(SOLUTION) − + = → = + 6 + 2 = 14 6 + 2(2 + 2) = 14 6 + 4 + 4 = 14 10 = 10 x = 1 (, ) y=2 1 +2 →4 CONCEPT SUMMARY: (continue) METHOD WHEN TO USE Elimination When the equations are in Ax +By = C form or the coefficients of one variable are the same and/or opposites 2 x 5 y 17 6 x 5 y 9 http://www.khanacademy.org/mat h/algebra/systems-of-eq-andineq/fast-systems-ofequations/v/solving-systems-ofequations-by-elimination http://player.discoveryeducation.com/index.cfm?guidAssetId=02B482AE-EB9F-4960-BC5C7D2360BDEE66 YOU TRY IT: Solve the system by Elimination: −2 + = 2 6 + 2 = 14 YOU TRY IT: (SOLUTION) − + = → −(− + = ) + = − = − + + = = 10 + =+ = + = = x=1 y=4 ADDITIONALLY: System of equations help us solve real world problems. http://player.discoveryeducation.com/index. cfm?guidAssetId=A9199767-40AB-4AD19493-9391E75638D0 VIDEO-Word Prob. NOTE: We can solve system of equations using a graph, the substitution or eliminations process. The best method to use will depend on the form of the equations and how precise we want the answer to be. 6.4 Application of Linear Systems: Break-Even Point: The point for business is where the income equals the expenses. GOAL: MODELING PROBLEMS: Systems of equations are useful to for solving and modeling problems that involve mixtures, rates and Break-Even points. Ex: A puzzle expert wrote a new sudoku puzzle book. His initial costs are $864. Binding and packaging each book costs $0.80. The price of the book is $2.00. How many books must be sold to break even? SOLUTION: 1) Write the system of equations described in the problem. Let x = number of books sold Let y = number of dollars of expense or income Expense: y = $0.80x + 864 Income: y = $2x SOLUTION: (Continue) 2) Solve the system of equations for the break-even point using the best method. To break even we want: Expense = Income $0.80x + 864 = $2x 864 = 2x -0.80x 864 = 1.2x 720 = x There should be 720 books sold for the puzzle expert to break-even. YOU TRY IT: Ex: A fashion designer makes and sells hats. The material for each hat costs $5.50. The hats sell for $12.50 each. The designer spends $1400 on advertising. How many hats must the designer sell to break-even? SOLUTION: 1) Write the system of equations described in the problem. Let x = number of hats sold Let y = number of dollars of expense or income Expense: y = $5.50x + $1400 Income: y = $12.50x SOLUTION: (Continue) 2) Solve the system of equations for the break-even point using the best method. To break even we want: Expense = Income $5.50x + $1400 = $12.50x 1400 = 12.5x -5.50x 1400 = 7x 200 = x There should be 200 hats sold for the fashion designer to break-even. VIDEOS: Special Linear Equations https://www.khanacademy.org/math/algebra/syst ems-of-eq-and-ineq/fast-systems-ofequations/v/special-types-of-linear-systems CLASSWORK: Page 386-388 Problems: As many as needed to master the concept. SUMMARY: http://www.bing.com/ videos/search?q=SYSTE M+OF+EQUATIONS+&v iew=detail&mid=2CFE6 3B47EDB353AFDCF2CF E63B47EDB353AFDCF& first=0&FORM=NVPFVR