Algebra II - Linear Equations in Three Variables - Quick Guides | Mathematics

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Study GuideAlgebra II–Linear EquaƟons in Three Variables1. Quiz: Linear EquaƟons: SoluƟons Using EliminaƟon with ThreeVariables1. QuestionSolve the system of equations:Answer Choicesx = 1; y = 3; z =−2x =−1; y =−3; z = 2x = 1; y =−3; z = 2Correct Answerx = 1; y = 3; z =−2Why This Is CorrectSubstitute the values into each equation to verify:1.2(1) + 3(3) + 4(−2)= 2 + 9−8= 3✔2.3(1)−4(3) + 2(−2)= 3−12−4=−13✔

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Study Guide3.1 + 5(3)−3(−2)= 1 + 15 + 6= 22✔Since the values satisfy all three equations, the correct solution isx = 1, y = 3, z =−2.2. QuestionSolve the system:Answer Choicesx = 2; y = 4; z = 5x =−2; y =−4; z = 5x =−2; y = 4; z =−5Correct Answerx = 2; y = 4; z = 5Why This Is CorrectCheck the values:1.2(2)−3(4)= 4−12=−83(5)−7 = 15−7 = 8Rearranging confirms the equality.2.4(5)−2(2) + 3(4)

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Study Guide= 20−4 + 12= 283.4(4) + 2(5)−3(2)= 16 + 10−6= 20The equations hold forx = 2, y = 4, z = 5, confirming the correct solution.3. QuestionSolve the system:Answer Choicesx = 5; y = 7; z =−6x =−5; y = 7; z =−6x = 5; y =−7; z = 6Correct Answerx = 5; y = 7; z =−6Why This Is CorrectSubstitute:1.3(7) + 2(5)−5(−6)= 21 + 10 + 30= 61✔2.4(−6) + 2(7) + 5

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Study Guide−24 + 14 + 5=−5✔3.3(5) + 2(7)−3(−6)= 15 + 14 + 18= 47✔All equations are satisfied, confirming the solution.4. QuestionSolve the system:Answer Choicesx = 3; y =−5; z =−4x =−3; y =−5; z = 4x =−3; y = 5; z =−4Correct Answerx = 3; y =−5; z =−4Why This Is CorrectSubstitute the values:1.6(−5) + 2(3) + 4(−4)=−30 + 6−16=−40 (matches the balanced equation form)2.5(−4) + 7(−5)−4(3)

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Study Guide=−20−35−12=−673.5(−5)−8(−4) + 6(3)=−25 + 32 + 18= 25These values satisfy the system, confirming the correct solution.5. QuestionSolve the system:Answer Choicesx =−4; y = 6; z = 12x =−4; y =−6; z = 12x = 4; y =−6; z =−12Correct Answerx =−4; y = 6; z = 12Why This Is CorrectSubstitute values:1.4(−4) + 3(6) + 3(12)=−16 + 18 + 36= 38 (matches equation balance)2.3(−4) + 7(6)

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Study Guide=−12 + 42= 302(12)−78 = 24−78 =−543.5(12) + 4(−4) + 4(6)= 60−16 + 24= 68Thusx =−4, y = 6, z = 12satisfies the system.2. Linear EquaƟons: SoluƟons Using EliminaƟon with Three VariablesWhen solving a system of equations with three variables, it’s a little more challenging than with twovariables, but you can still use the elimination method, which works like the one used for twovariables. There’s also another method using a 3x3 matrix,but let’s focus on elimination for now.Steps for Elimination with Three Variables:1.Write the equations in standard form: Make sure the equations don’t have any decimals orfractions.2.Choose a variable to eliminate: Pick one of the variables (like x, y, or z) to eliminate, andchoose two equations to use.3.Eliminate the variable: Use the chosen two equations to eliminate the variable you picked inStep 2.4.Solve for the other two variables: After eliminating one variable, you'll be left with twoequations and two variables. Solve them to find values for the remaining variables.5.Substitute the values back: Take the values you found in Step 4 and put them back into anyof the original equations to find the last variable.6.Check your work: Double-check by plugging all your answers into the original equations tomake sure everything adds up.

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Study GuideExample 1:Solving with EliminationConsider this system of equations:Step 1: Choose a variable to eliminateLet's choosexto eliminate. We’ll use equations (1) and (2):This gives us the new equation:Now, adding these two equations:The result is:Step 2: Choose two more equationsNext, let's eliminatexagain but use equations (2) and (3). After multiplying the appropriate terms,you'll get the new equation:

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Study GuideStep 3: Solve the system of equations (4) and (5)Now solve the two equations:By solving, you findz = 3.Step 4: Substitute back to find y and xNow substitutez = 3into one of the earlier equations (like equation (4)) to findy = 2. Then use thevalue ofyandzin equation (2) to findx =-1.Step 5: Final checkSubstitutex =-1,y = 2, andz = 3into all original equations to ensure they are correct.The solution is:Example 2:Another Set of EquationsGiven this system of equations:Step 1: Eliminate yFirst, use equations (2) and (3) to eliminatey. After manipulating these, substitute the results to solveforxandz.Step 2: Final SubstitutionOncex = 4andz = 3are found, substitute these into the remaining equations to findy =-2.

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