Algebra II – Quadratic Systems | Mathematics

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Study GuideAlgebra II–Quadratic Systems1.Systems of Equations Solved AlgebraicallyWhat Is a System of Equations?Asystem of equationsis a set of two or more equations that contain thesame variables.The goal is to find values of the variables thatsatisfy all equations at the same time.For example:A solution to a system is usually written as anordered pair((x,y)).Methods for Solving Systems AlgebraicallyThere are two main methods:1.Substitution Method2.Elimination MethodBoth methods aim toremove one variableso the system becomes easier to solve.Example 1—Elimination MethodSolve the system:Step 1: Multiply the second equation by 2

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Study GuideStep 2: Add the equationsAddthem:Step 3: Substitute into the first equationUsingCase 1: (x = 2)Solutions:

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Study GuideCase 2: (x =-2)Solutions:Final SolutionsThe system hasfour solutions:These represent thepoints where the graphs intersect.Example 2—Substitution MethodSolve:

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Study GuideStep 1: Solve the second equation for xStep 2: Substitute into the first equationReplace (x):Step 3: ExpandStep 4: Move everything to one sideFactor:Step 5: Solve for y

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Study GuideorStep 6: Find xUsing (x = y + 2)If (y = 6)If (y =-8)Final SolutionsThese are thepoints where the graphs intersect.Key Ideas to RememberElimination Method1.Multiply equations if needed.2.Add or subtract equations to eliminate one variable.3.Solve the remaining equation.4.Substitute back to find the other variable.Substitution Method

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Study Guide1.Solve one equation for a variable.2.Substitute into the other equation.3.Solve the resulting equation.4.Substitute back to find theremaining variable.SummaryAsystem of equationsinvolves two or more equations with the same variables.Two common algebraic methods are:SubstitutionSolve one equation for a variable and substitute into the other.EliminationAdd or subtract equations to eliminate a variable.Thesolutions represent the intersection points of the graphsof the equations.2.Quiz: Systems of Equations Solved Algebraically1. QuestionSolve the system:Answer Choices•(x =-2, y = 2)•(x = 2, y =-2)•(x = 3, y=-2)

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Study GuideCorrect AnswerWhy This Is CorrectFrom the second equation:Substitute into the first equation:Solving gives:Substitute back:2. QuestionSolve the system:Answer Choices•(x = ±3, y = ±4)•(x = ±4, y = ±3)•(x = ±3, y = ±2)Correct Answer

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Study GuideWhy This Is CorrectLet:Then:Solve the system:Thus:3. QuestionSolve the system:Answer Choices•(a = ±2, b = ±1)•(a = 2, b = ±3)•(a = ±1, b = ±2)Correct AnswerWhy This Is Correct

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Study GuideLet:Then:Solving gives:So:4. QuestionSolve the system:Answer Choices•(p = ±3, q = ±4)•(p = ±2, q = ±5)•(p = ±5, q = ±2)Correct AnswerWhy This Is CorrectRewrite equations:

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