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## SYSTEMS of EQUATIONS

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**SYSTEMS of EQUATIONS**MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION A SYSTEM of EQUATIONS is a number of equations containing the same variables MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION You and a friend go to get a ‘burger… You order 2 ‘burgers and a coke… $4.25 Your friend orders 3 ‘burgers and 2 cokes What is the cost of 1 ‘burger? $7.00 What is the cost of 1 coke? This is a SYSTEM of EQUATIONS MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION 2 ‘burgers and a coke……$4.25 3 ‘burgers and 2 cokes…….$7.00 We have 2 equations We have 2 variables X= cost of a burger Y= cost of a coke MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION This is our SYSTEM of EQUATIONS We can solve this system many ways: • GRAPHING • CRAMER’s RULE • SUBSTITUTION • INVERSE MATRICES • GAUSSIAN ELIMINATION • CHOLESKY DECOMPOSITION • To name a few! We will work with these methods. MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION This is our SYSTEM of EQUATIONS The SOLUTION is the value of both x and y. In this case, a burger costs $1.50, and a coke costs $1.25 MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION Solving a system of equations by GRAPHING: Just graph both of the equations, and find the POINT of INTERSECTION The solution is That means the x value is 3 and the y value is 1 MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION Solving a system of equations by SUBSTITUTION: This is where we plug one entire equation into another. MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION Solving a system of equations by ELIMINATION: This is where we multiply, and add or subtract the equations to solve them. MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION Sometimes it is going to turn out that there is NO SOLUTION or MANY SOLUTIONS In our burger problem, NO SOLUTION: means you and your friend were not charged the same prices MANY SOLUTIONS: means you and your friend ordered the same meal MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION It can get much trickier if we add a third variable, like fries! If there are 3 unknowns (burger, coke, fries) then you would need a third person ordering a different combination However many variables you need to find… You need the same number of unique equations MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION Systems of equations and multiple choice: Which of the following is a solution to the system of equations? A) (3,4) B) (4, -3)C) (3, -4) D) (4, 3) E) (-3, -4) MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION Example Determine whether the given point is a solution of the following system. point: (– 3, 1) system: x – y = – 4 and 2x + 10y = 4 • Plug the values into the equations. First equation: – 3 – 1 = – 4 true Second equation: 2(– 3) + 10(1) = – 6 + 10 = 4 true • Since the point (– 3, 1) produces a true statement in both equations, it is a solution. MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**INTRODUCTION Example Determine whether the given point is a solution of the following system point: (4, 2) system: 2x – 5y = – 2 and 3x + 4y = 4 Plug the values into the equations First equation: 2(4) – 5(2) = 8 – 10 = – 2 true Second equation: 3(4) + 4(2) = 12 + 8 = 20 4 false Since the point (4, 2) produces a true statement in only one equation, it is NOT a solution. MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SOLVE by GRAPHING SOLVE A SYSTEM OF EQUATIONS BY GRAPHING MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SOLVE by GRAPHING If Dave is driving on this street Solving a system of equations by graphing is basically the same as answering this question: And Jessica is driving on this street What is the only intersection BOTH cars will drive by? MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SOLVE by GRAPHING What is the solution to this system of equations? MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SOLVE by GRAPHING What are the solution to these systems of equations? MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SOLVE by GRAPHING What are the solution to these systems of equations? MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**y x SOLVE by GRAPHING What is the solution to this system of equations? These lines are parallel. There is no intersection NO SOLUTION MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SOLVE by GRAPHING What is the solution to this system of equations? y When lines PERFECTLY overlap like this, They are the same line x INFINITELY MANY SOLUTIONS MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**y (-5, 5) (6, 6) (-2, 4) (4, 2) (1, 3) x (3, 0) (0, -6) SOLVE by GRAPHING Solve this system of equations by graphing: The lines APPEAR to intersect at (4, 2). CHECK: First equation, 2(4) – 2 = 8 – 2 = 6 true Second equation, 4 + 3(2) = 4 + 6 = 10 true MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**y (2, 0) (-1, 3) (-1, -1) (2, 5) (0, 3) x SOLVE by GRAPHING Solve this system of equations by graphing: Where do they intersect? It cannot be determined from the graph MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**y (6, 4) (0, 2) (6, 1) (3, 0) x (-6, 0) (0, -1) SOLVE by GRAPHING Solve this system of equations by graphing: NO SOLUTION MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SOLVE by GRAPHING How many solutions does each system of equations have? MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**• NUMBER OF SOLUTIONS OF A LINEAR SYSTEM y y y x x x SOLVE by GRAPHING Lines intersect one solution Lines are parallel no solution Lines coincide infinitely many solutions MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**Graphs that don’t give us solutions SOLVE by GRAPHING • These lines don’t touch • BUT… these lines are not parallel • There is a solution, but we can’t tell from this graph MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SOLVE by GRAPHING SOLVING A SYSTEM of EQUATIONS by GRAPHING GOOD BAD Easy Graphing can take time Inaccurate MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SUBSTITUTION Solving by SUBSTITUTION is based on this kind of problem: Evaluate the equation if Except, instead of putting just a number in for x, we are going to put in a whole equation! MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SUBSTITUTION Solve this system by SUBSTITUTION Make sure to include these parenthesis But now we only know x. We have to plug it back in to one of the equations to find y. MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SUBSTITUTION Solve this system by SUBSTITUTION MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SUBSTITUTION Solve this system by SUBSTITUTION Practice #2 MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SUBSTITUTION Solve this system by SUBSTITUTION Practice #3 MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SUBSTITUTION Special solutions from substitution: All the variables have cancelled out. When this happens the answer will be either NO SOLUTION or MANY SOLUTIONS What we are left with is TRUE INFINITELY MANY SOLUTIONS MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SUBSTITUTION Special solutions from substitution: All the variables have cancelled out. When this happens the answer will be either NO SOLUTION or MANY SOLUTIONS What we are left with is FALSE NO SOLUTION MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SUBSTITUTION Here’s the rule: If you are solving an equation, and the X’s & Y’s disappear, If what is left is FALSE If what is left is TRUE ∞ solutions NO SOLUTION MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION GREASY BOLAN’s DINER Happy Meal #1 3 Burgers and a coke ……………… $5.42 Happy Meal #2 2 Burgers and a coke ……………… $3.92 MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION GREASY BOLAN’s DINER Happy Meal #3 2 Chicken Sandwiches and a shake ……………… $5.20 Happy Meal #4 5 Chicken Sandwiches and 2 shakes ….………… $12.45 MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION GREASY BOLAN’s DINER Happy Meal #5 1 Burrito and 1 Chocolate Milk …….…………… $2.85 Happy Meal #6 5 Burritos and 3 Chocolate Milks ……………… $13.55 MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION GREASY BOLAN’s DINER Happy Meal #7 1 chicken nugget and 1 root beer…….………………… $0.95 Happy Meal #8 10 chicken nuggets and 10 root beers…….…………… $9.50 MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION Linear Combinations MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION Linear Combinations MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION Sometimes when you solve a system of equations, all the variables cancel out: FALSE NO SOLUTION TRUE ∞ SOLUTIONS MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**LINEAR COMBINATION ∞ SOLUTIONS No SOLUTION Even though we get 0, this is still a normal problem, so we just continue to solve MENU OPENERS ASSIGNMENTS REVIEWS Activities**SYSTEMS of EQUATIONS**SPECIAL SYSTEMS MENU OPENERS ASSIGNMENTS REVIEWS Activities