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4. Solving Systems Graphically The Substitution Method The Elimination Method Applications of Systems of Equations Systems of Equations in Two Variables Systems of Equations and Matrices > Systems of Equations in Two Variables Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_40&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. The graphical method is a great way to solve a system of equations, and also to check your work if you are solving the system using elimination or substitution. There are many ways to write any equation, but if you are going to solve the system graphically, it is helpful to first isolate the y term on one side of the equation/s. You can solve the system by locating the intersections between the different equations in the system. It is possible to have more than one answer that satisfies all equations in a system. Solving Systems Graphically Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-equations-in-two-variables- 40/solving-systems-graphically-190- 5863?campaign_content=book_196_section_40&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess System of Equations with multiple answers View on Boundless.com Systems of Equations and Matrices > Systems of Equations in Two Variables

6. A system of equations is a set of equations that can be solved using a particular set of values. The substitution method works by expressing one of the variables in terms of another, then substituting it back into the original equation thus simplifying it. It is very important to check your work once you have found a set of values for the variables. Do this by substituting the values you found back into the original equations. The answer to the system of equations can be written as an ordered pair (x,y). The Substitution Method Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-equations-in-two-variables- 40/the-substitution-method-191- 5015?campaign_content=book_196_section_40&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Systems of Equations and Matrices > Systems of Equations in Two Variables

7. In order to easily solve the system of equations, it helps to first set the equations up in similar way. For example, x+y=-1 and 2y+x=-4 should be written: x+ y=-1 and x+2y=-4. Once, the values for the remaining variables have been found successfully, then go back and plug that result into one of the original equation and find the correct value for the other variable. Always check the work. This is done by plugging both values into one or both of the original equations. The Elimination Method Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-equations-in-two-variables- 40/the-elimination-method-192- 4336?campaign_content=book_196_section_40&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Systems of Equations and Matrices > Systems of Equations in Two Variables

8. If you have a problem that includes multiple variables, you can solve it by creating a system of equations. The first step to solving a multivariate problem is to identify and label the variables. Once variables are defined, determine the relationships between them and write them as equations. Applications of Systems of Equations Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-equations-in-two-variables- 40/applications-of-systems-of-equations-193- 2244?campaign_content=book_196_section_40&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Pendulum View on Boundless.com Systems of Equations and Matrices > Systems of Equations in Two Variables

9. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

10. Key terms constraint A condition that a solution to a problem must satisfy. elimination method Method of solving a system of equations by eliminating one variable in order to more simply solve for the remaining variable. ordered pair A set containing exactly two elements in a fixed order, used to represent a point in a Cartesian coordinate system. Notation: (x, y). substitution method Method of solving a system of equations by putting the equation in terms of only one variable system of equations A set of equations with multiple variables which can be solved using a specific set of values. The graphical method A way of visually finding a set of values that solves a system of equations. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

11. Interactive Graph: System of Equations with Two Variables Graph of a systems of equations with two variables, with the equations and. This figure is a graphical representation of the example equations. You can use this to check your answers, and see that the ordered pair you come up with matches where the lines intersect with one another. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-78fec697-5f95-48e4-8b1f-a61cfaa0cb98 View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-78fec697-5f95-48e4-8b1f-a61cfaa0cb98View on Boundless.com Systems of Equations and Matrices

12. System of linear equations with two variables This graph shows a system of equations with two variables with only one set of answers. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikibooks. "Two linear equation add method." CC BY-SA 3.0 http://en.wikibooks.org/wiki/File:Two_linear_equation_add_method.PNG View on Boundless.comCC BY-SA 3.0http://en.wikibooks.org/wiki/File:Two_linear_equation_add_method.PNGView on Boundless.com Systems of Equations and Matrices

13. System of Equations with multiple answers This is an example of a system of equations shown graphically that has two sets of answers that will satisfy both equations in the system. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Simultaneous equations example 1." CC BY-SA http://en.wikipedia.org/wiki/File:Simultaneous_equations_example_1.svg View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:Simultaneous_equations_example_1.svgView on Boundless.com Systems of Equations and Matrices

14. Pendulum This animation shows the velocity and acceleration vectors for a pendulum. One may note that at the maximum height of the pendulum's mass, the velocity is zero. This corresponds to zero kinetic energy and thus all of the energy of the pendulum is in the form of potential energy. When the pendulum's mass is at its lowest point, all of its energy is in the form of kinetic energy and we see its velocity vector has a maximum magnitude here. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Pendulum." CC BY http://en.wikipedia.org/wiki/Pendulum View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/PendulumView on Boundless.com Systems of Equations and Matrices

15. Interactive Graph: Graphical Representation of a Systems of Equations Graph of systems of equations with the equations and Once you have created your system of equations, one way to solve it is by showing them graphically, and then finding the intersecting point or points. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graphical Representation of a Systems of Equations." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graphical- representation-of-a-systems-of-equations View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graphical- representation-of-a-systems-of-equationsView on Boundless.com Systems of Equations and Matrices

16. Interactive Graph: System of Equations with Two Variables Graph of system of equations with two variables with the equations and. This graph demonstrates that the system of equations has a specific set of values which will solve BOTH equations. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph:." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-8a5a3300-0928-48de-b7cf-89be47a83aab View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-8a5a3300-0928-48de-b7cf-89be47a83aabView on Boundless.com Systems of Equations and Matrices

17. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Solve the following system of equations using substitution: 3y- 2x=11 and y+2x=9. A) x=2 and y=5 B) x=5 and y=2 C) x=9 and y=11 D) x=3 and y=7

18. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices Solve the following system of equations using substitution: 3y- 2x=11 and y+2x=9. A) x=2 and y=5 B) x=5 and y=2 C) x=9 and y=11 D) x=3 and y=7

19. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices What could we substitute in for y in the following system of equations: 2x-3y=-2 and 4x+y=24? A) y=-4x+24 B) y=4x+24 C) y=2x+2/3 D) y=2x+3

20. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices What could we substitute in for y in the following system of equations: 2x-3y=-2 and 4x+y=24? A) y=-4x+24 B) y=4x+24 C) y=2x+2/3 D) y=2x+3

21. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices At a local animal shelter, the supplies for each dog costs twice as much as the supplies for a cat. We need to feed 164 cats and 24 dogs, with a budget of $4240. How much can we spend on each dog? A) $20 B) $60 C) $40 D) $50

22. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices At a local animal shelter, the supplies for each dog costs twice as much as the supplies for a cat. We need to feed 164 cats and 24 dogs, with a budget of $4240. How much can we spend on each dog? A) $20 B) $60 C) $40 D) $50

23. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Ben rides the Ferris wheel 3 times and goes down the water slide 3 times. Jeff also rides the Ferris wheel 3 times but only goes down the water slide twice. Ben spends $17.70 and Jeff spends $15.55. How much does the water slide cost per ride? A) $3.75 B) $2.15 C) $2.75 D) $3.15

24. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices Ben rides the Ferris wheel 3 times and goes down the water slide 3 times. Jeff also rides the Ferris wheel 3 times but only goes down the water slide twice. Ben spends $17.70 and Jeff spends $15.55. How much does the water slide cost per ride? A) $3.75 B) $2.15 C) $2.75 D) $3.15

25. Attribution OER Commons. CC BY http://www.oercommons.org/courses/simultaneous-equations-homework-simultaneous-equations/viewCC BYhttp://www.oercommons.org/courses/simultaneous-equations-homework-simultaneous-equations/view Wikipedia. "System of equations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/System_of_equationsCC BY-SA 3.0http://en.wikipedia.org/wiki/System_of_equations Wiktionary. "constraint." CC BY-SA 3.0 http://en.wiktionary.org/wiki/constraintCC BY-SA 3.0http://en.wiktionary.org/wiki/constraint Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/ordered-pairCC BY-SA 3.0http://www.boundless.com//algebra/definition/ordered-pair Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/system-of-equationsCC BY-SA 3.0http://www.boundless.com//algebra/definition/system-of-equations Wikipedia. "Systems of equations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Systems_of_equationsCC BY-SA 3.0http://en.wikipedia.org/wiki/Systems_of_equations Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/system-of-equationsCC BY-SA 3.0http://www.boundless.com//algebra/definition/system-of-equations Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/elimination-methodCC BY-SA 3.0http://www.boundless.com//algebra/definition/elimination-method Wikibooks. "Algebra/Systems of Equations." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Systems_of_EquationsCC BY-SA 3.0http://en.wikibooks.org/wiki/Algebra/Systems_of_Equations Wikipedia. "Systems of equations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Systems_of_equationsCC BY-SA 3.0http://en.wikipedia.org/wiki/Systems_of_equations Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/ordered-pairCC BY-SA 3.0http://www.boundless.com//algebra/definition/ordered-pair Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/substitution-methodCC BY-SA 3.0http://www.boundless.com//algebra/definition/substitution-method Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/system-of-equationsCC BY-SA 3.0http://www.boundless.com//algebra/definition/system-of-equations Wikipedia. "Systems of equations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Systems_of_equationsCC BY-SA 3.0http://en.wikipedia.org/wiki/Systems_of_equations Wikipedia. "Linear equation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Linear_equation#Linear_equations_in_two_variablesCC BY-SA 3.0http://en.wikipedia.org/wiki/Linear_equation#Linear_equations_in_two_variables Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/system-of-equationsCC BY-SA 3.0http://www.boundless.com//algebra/definition/system-of-equations Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/the-graphical-methodCC BY-SA 3.0http://www.boundless.com//algebra/definition/the-graphical-method Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices