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Fontana Unified School DistrictEvery Student Successful Engaging Schools Empowered CommunitiesOffline Distance LearningSecondaryAdvanced Math 7May 2020School Name:Student ID#:Math Teacher Name:Period:May 2020

May 2020

Advanced Math 7: May 4 – May 8Concept: Solving Systems by GraphingSystems of linear equations- A set of two or more linear equations with the samevariablesSolutions to systems of linear equations- A solution to a system of equations is anordered pair that is true for both equations-When you graph a system, the solution iswhere the two lines intersectOften, the system will be marked with bracketslike above left example, but it is not required.-When you graph a linear system, you graph bothlines on the same coordinate plane.One SolutionInfinite SolutionsNo Solutions-The system will have onesolution when the lines crossexactly one time.-The system will have aninfinite number of solutionswhen the lines are the same,or they touch at every point.-The system will have nosolutions if the lines neverintersect at any point.-Here, the solution is (15, 9)because that is where the linesintersect.-These lines fall on top of eachother, so there is an infinitenumber of solutions.These lines will have thesame slope and the same yintercept.-These lines will neverintersect, so they have nosolution.These lines will have thesame slope but different yintercepts.-Ex.𝑦𝑦 2π‘₯π‘₯ 2𝑦𝑦 2π‘₯π‘₯ 2-Ex.𝑦𝑦 3π‘₯π‘₯𝑦𝑦 3π‘₯π‘₯ 6May 2020

Checking SolutionsChecking Solutions for a System--Substitute the coordinate pair the equation, if it istrue, then it is a solutionSolving by GraphingIf the coordinate pair is a solution to a systemof equations, then it must be true for BOTHequationsSolving by Graphing: Not in π’šπ’š π’Žπ’Žπ’Žπ’Ž 𝒃𝒃1) Find the slope and y-intercept1) The first thing you need to is solve eachequation for y.2) Graph both lines and identify where theyintersect.2) Then it’s the same steps as before – grapheach line and determine whether theyintersect.3) Check your solution for both equations3) Check each solution to determine if it istrue.May 2020

Problems:1. Determine if (3, 4) is a solution to the system.2π‘₯π‘₯ 𝑦𝑦 2π‘₯π‘₯ 2𝑦𝑦 113. Identify the solution to the system.The solution is ( , ).5. How many solutions does these systems ofequations have? Justify how you know.2. Determine if (3, -1) is a solution to thesystem.𝑦𝑦 3π‘₯π‘₯ 1𝑦𝑦 2π‘₯π‘₯4. Identify the solution to the system.The solution is ( , ).6. Describe the difference in the number ofsolutions between a system of linearequations that coincide and a system of linearequations that are parallel.May 2020

7. Graph the system and identify the solution.8. Graph the system and identify the solution.𝑦𝑦 3π‘₯π‘₯ 2𝑦𝑦 2π‘₯π‘₯ 32π‘₯π‘₯ 2𝑦𝑦 64π‘₯π‘₯ 6𝑦𝑦 129. Graph the system and identify the solution.10. Graph the system and identify the solution.𝑦𝑦 3π‘₯π‘₯ 2𝑦𝑦 3π‘₯π‘₯ 2𝑦𝑦 2π‘₯π‘₯ 12𝑦𝑦 4π‘₯π‘₯ 2May 2020

Advanced Math 7: May 11 – May 15Concept: Solving Systems AlgebraicallySolving Systems AlgebraicallySubstitution---Solving a system by substitution is where yousubstitute a value or expression from the firstequation into the second equation to solve for oneof the variables.You need to have one equation by in β€œy ” or β€œx ” form.--𝑦𝑦 3π‘₯π‘₯ 42π‘₯π‘₯ 3𝑦𝑦 7The first equation is set up as y , which meanswe can substitute the 3x - 4Substitution Example-Elimination2π‘₯π‘₯ 𝑦𝑦 11𝑦𝑦 3π‘₯π‘₯ 9The second equation has a y , so I take the 3x –9 and substitute it for the y in the first equation.2π‘₯π‘₯ ( ) 11---Now I solve for x.2π‘₯π‘₯ 3π‘₯π‘₯ 9 115π‘₯π‘₯ 9 115π‘₯π‘₯ 20π‘₯π‘₯ 4Substitute the x back into any equation. I’mchoosing the second one because it seems like itwill be easier to find y.𝑦𝑦 3(4) 9𝑦𝑦 12 9𝑦𝑦 32π‘₯π‘₯ 3𝑦𝑦 12π‘₯π‘₯ 3𝑦𝑦 15We have -3y in the first equation and 3y in thesecond equation, so they add up to 0Elimination Example-2π‘₯π‘₯ 3𝑦𝑦 20 2π‘₯π‘₯ 𝑦𝑦 4I see that I have a -2x and 2x in my twoequations, so that is the variable I will β€œeliminate”or make 0.2π‘₯π‘₯ 3𝑦𝑦 20 2π‘₯π‘₯ 𝑦𝑦 4𝑦𝑦 3π‘₯π‘₯ 92π‘₯π‘₯ (3π‘₯π‘₯ 9) 11Solving a system by elimination is where you lineup both equations and add or subtract themtogether to eliminate one variable completely (itadds up to 0), allowing you to solve for the othervariable.You need one set of coefficients to be opposite ofeach other ( -4 and 4 or -1 and 1)--0 4𝑦𝑦 24From there, I solve for y.4𝑦𝑦 24𝑦𝑦 6I substitute the 6 back into either equation tosolve for x. I’m choosing the first one.2π‘₯π‘₯ 3(6) 202π‘₯π‘₯ 18 202π‘₯π‘₯ 2π‘₯π‘₯ 1The solution to the system is (1, 6).The solution to the system is (4, 3).May 2020

Setting a Problem up for SubstitutionSetting a Problem up for Elimination--If a problem isn’t set up for substitution, movethings around until you have anπ‘₯π‘₯ π‘œπ‘œπ‘œπ‘œ 𝑦𝑦 If there aren’t any coefficients that are alreadyopposites, you can multiple one entire equation tocreate opposites.(3, 2)(2, 2)The Variable Disappeared!-There are two situations where we get an answer that makes the algebra look strange, meaning thevariable disappears completely or you end up with something like x x.-When we solve for an infinite number of solutions, it will look like this:1 1-When we solve for no solutions, it will look like this:1 2Infinite SolutionsNo Solutions-We know what the graph looks like, now we willlearn what the algebra looks like.-We know what the graph looks like, now we willlearn what the algebra looks like.-I’ve substituted an expression and I get a problemthat looks like this:-I’ve substituted an expression and I get a problemthat looks like this:--2π‘₯π‘₯ 3 2π‘₯π‘₯ 3After subtracting 2x from each side, I get this:3 3This is true. This is always true. So, this systemwill have an infinite number of solutions.2π‘₯π‘₯ 1 2π‘₯π‘₯ 3-After subtracting 2x from each side, I get this:-1 3This is not true. This is never true. So, thissystem will have no solutions.May 2020

Infinite Solutions Example𝑦𝑦 2π‘₯π‘₯ 32𝑦𝑦 4π‘₯π‘₯ 62(2π‘₯π‘₯ 3) 4π‘₯π‘₯ 64π‘₯π‘₯ 6 4π‘₯π‘₯ 6 6 6There is an infinite number of solutions.No Solutions Example2π‘₯π‘₯ 3𝑦𝑦 72π‘₯π‘₯ 3𝑦𝑦 3I subtract the two equations to eliminate the variables0 40 does not equal 4. There are no solutions.Recap:May 2020

May 2020

Problems:1. Some systems of two linear equations in twovariables have a single solution (x, y). Othershave no solutions, while still others have aninfinite number of solutions.2. When solving systems algebraically, howmany solutions will each system have when Isolve for a variable and see this:Match each description to number of solutions.x 1Parallel lines2 2Intersecting linesCoinciding lines- Singlesolution-1 5- No solution-Infinitesolution3. Using substitution, solve the system.π‘₯π‘₯ 2𝑦𝑦 18π‘₯π‘₯ 4𝑦𝑦5. Using elimination, solve the system.5π‘₯π‘₯ 2𝑦𝑦 26 π‘₯π‘₯ 2𝑦𝑦 224. Using substitution, solve the system.π‘₯π‘₯ 3𝑦𝑦 22π‘₯π‘₯ 𝑦𝑦 106. Using elimination, solve the system.6π‘₯π‘₯ 6𝑦𝑦 283π‘₯π‘₯ 3𝑦𝑦 14May 2020

7. Which method makes more sense to solve thissystem? Why?8. Solve the system with your chosen method.9. Which method makes more sense to solve thissystem? Why?𝑦𝑦 4π‘₯π‘₯𝑦𝑦 4π‘₯π‘₯ 710. Solve the system with your chosen method.𝑦𝑦 4π‘₯π‘₯𝑦𝑦 4π‘₯π‘₯ 711. Solve the system with any method.12. Solve the system with any method.π‘₯π‘₯ 𝑦𝑦 143π‘₯π‘₯ 𝑦𝑦 36May 2020

Infinite Solutions - The system will have an infinite number of solutions when the lines are the same, or they touch at every point. - These lines fall on top of each other, so there is an infinite number of solutions. - These lines will have the sam

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