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Algebra II - Linear Sentences in Two Variables

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Study GuideAlgebra II–Linear Sentences in Two Variables1. Linear EquaƟons: SoluƟons Using Graphing with Two VariablesSometimes you are giventwo linear equationsand asked to find a solution that satisfiesbothequations at the same time.One way to do this isgraphing.When you graph both equations on thesame coordinate plane, the point where the two linesintersect (cross)represents thesolution to the system.That intersection point gives theordered pair (x, y)that works for both equations.Example 1:SolveFigure 1Solve the system by graphing:

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Study GuideStep 1: Graph Both EquationsPlot each equation as a line on thesame coordinate plane.•The first equation forms one line.•The second equation forms another line.Step 2: Find the IntersectionLook for the point where thetwo lines cross.From the graph, the lines intersect at:So thesolution to the system is:Step 3: Check the SolutionTo make sure the solution is correct, substitute (x = 3) and (y =-2) into both equations.Check the First EquationSubstitute the values:The equation is true✔

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Study GuideCheck the Second EquationSubstitute the values:This equation is also true✔Since the ordered pair works forboth equations, it is thecorrect solution.Important Note About GraphingGraphing is helpful for understanding systems visually, but it has some limitations.•The intersection point must beeasy to see on the graph.•It usually works best when the solution isclose to the originand hasinteger values.•Sometimes the exact intersection is hard to estimate just by looking at the graph.Because of this,graphing is often used less frequently than other methods(like substitution orelimination) when solving systems.Special Types of SystemsWhen graphing systems of equations, two special situations can occur.1. Dependent SystemAdependent systemhappens when the two equations representthe same line.This means the graphscompletely overlap (coincide).Since every point on the line satisfies both equations, the system hasinfinitely many solutions.In other words, the solution set can be written asall points on the line.

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Study Guide2. Inconsistent SystemAninconsistent systemhappens when the two lines areparallel.Parallel linesnever intersect, so there isno solution.The solution set is written as:This symbol means theempty set, which meansno solutions exist.Summary•Asystem of equationscontains two or more equations with the same variables.•Graphingboth equations on the same coordinate plane helps find the solution.•Theintersection point of the two linesis the solution.•Alwayscheck the solutionby substituting the values into both equations.Special Cases•Dependent system:Same line→infinitely many solutions•Inconsistent system:Parallel lines→no solutionExample solution from the graph:This point satisfies both equations.2. Quiz: Linear EquaƟons: SoluƟons Using Graphing with Two Variables1.QuestionSolve the system of equations:

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Study GuideAnswer Choices•(x =-3; y = 5)•(x = 3;y =-5)•(x = 5;y =-3)Correct Answer(x = 3;y =-5)Why This Is CorrectWe solve the system using theelimination method.1.Multiply the first equation by2so the (x) coefficients match:2. Subtract the second equation:3. Solve for (y):4. Substitute (y =-5) into (6x + 3y = 3):Therefore, the solution to the system is:

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Study Guide2. QuestionSolve the system of equations:Answer Choices•(x = 2;y = 7)•(x =-2;y = 7)•(x = 3;y = 6)Correct Answer(x = 2;y = 7)Why This Is CorrectFirst solve the second equation for (y):Next substitute this expression for (y) into the first equation:

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Study GuideNow substitute (x = 2) back into (y = 17–5x):Therefore, the solution to the system is:3. QuestionSolve the system of equations:Answer Choices(x = 2; y =-3)(x =-2; y = 3)(x =-2; y =-3)Correct Answer(x = 2; y =-3)Why This Is CorrectUse theelimination method.1.Multiply the first equation by7and the second equation by2to align the (x) terms:

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Study Guide2.Subtract the second equation from the first:3.Solve for (y):4.Substitute (y =-3) into the first equation:But check with the second equation:This satisfies the equation, so the correct solution is:Therefore, the correct answer is:(x =-2; y =-3)4. QuestionSolve the system of equations:

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Study GuideAnswer Choices•(x =-3; y = 7)•(x = 3; y =-7)•(x = 7; y =-3)Correct Answer(x =-3;y = 7)Why This Is CorrectFirst rewrite the second equation in standard form:Now eliminate (x). Multiply the second equation by5and the first equation by2:Add the equations:Substitute (y = 7) into the first equation:Therefore, the solution to the system is:

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Study Guide5. QuestionSolve the system of equations:Answer Choices•(x = 3; y =-5)•(x = 4; y =-4)•(x =-3; y = 5)Correct Answer(x = 4;y =-4)Why This Is CorrectFirst rewrite the second equation in standard form.Add (2y) to both sides:Now we have the system:Add the equations to eliminate (y):
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Document Details

University
Texas A&M University
Course
Algebra II
Subject
Mathematics

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