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tutionWarm UpLesson PresentationLesson QuizHoltHoltMcDougalAlgebra 1AlgebraAlgebra11HoltMcDougal
Solving Systems by SubstitutionWarm UpSolve systems of linear equations for x.1. y x 3x y–32. y 3x – 4Simplify each expression.3. 2(x – 5)2x – 104. 12 – 3(x 1)Holt McDougal Algebra 19 – 3x
Solving Systems by SubstitutionWarm Up ContinuedEvaluate each expression for the givenvalue of x.5.x 8 for x 6 126. 3(x – 7) for x 10Holt McDougal Algebra 19
Solving Systems by SubstitutionObjectiveSolve systems of linear equations in twovariables by substitution.Holt McDougal Algebra 1
Solving Systems by SubstitutionSometimes it is difficult to identify theexact solution to a system by graphing. Inthis case, you can use a method calledsubstitution.The goal when using substitution is toreduce the system to one equation thathas only one variable. Then you cansolve this equation by the methodstaught in Chapter 2.Holt McDougal Algebra 1
Solving Systems by SubstitutionSolving Systems of Equations by SubstitutionStep 1Solve for one variable in at least oneequation, if necessary.Step 2Substitute the resulting expression into theother equation.Step 3Solve that equation to get the value of thefirst variable.Step 4Substitute that value into one of the originalequations and solve.Step 5Write the values from steps 3 and 4 as anordered pair, (x, y), and check.Holt McDougal Algebra 1
Solving Systems by SubstitutionExample 1A: Solving a System of Linear Equations bySubstitutionSolve the system by substitution.y 3xy x–2Step 1 y 3xy x–2Both equations are solved for y.Step 2Substitute 3x for y in the secondequation.Solve for x. Subtract x from bothsides and then divide by 2.y x–23x x – 2Step 3 –x–x2x –22x –222x –1Holt McDougal Algebra 1
Solving Systems by SubstitutionExample 1A ContinuedSolve the system by substitution.Step 4Step 5Write one of the originalequations.Substitute –1 for x.y 3xy 3(–1)y –3(–1, –3)Write the solution as anordered pair.Check Substitute (–1, –3) into both equations in thesystem.y 3xy x–2–3 3(–1)–3 –1 – 2–3Holt McDougal Algebra 1–3 –3–3
Solving Systems by SubstitutionExample 1B: Solving a System of Linear Equations bySubstitutionSolve the system by substitution.y x 14x y 6Step 1 y x 1The first equation is solved for y.Step 2 4x y 6Substitute x 1 for y in the4x (x 1) 6second equation.Simplify. Solve for x.5x 1 6Step 3–1 –1Subtract 1 from both sides.5x 55x 5Divide both sides by 5.55x 1Holt McDougal Algebra 1
Solving Systems by SubstitutionExample1B ContinuedSolve the system by substitution.Step 4y x 1Step 5y 1 1y 2(1, 2)Write one of the originalequations.Substitute 1 for x.Write the solution as anordered pair.Check Substitute (1, 2) into both equations in thesystem.y x 14x y 62 1 14(1) 2 62 2 6 6 Holt McDougal Algebra 1
Solving Systems by SubstitutionExample 1C: Solving a System of Linear Equations bySubstitutionSolve the system by substitution.x 2y –1x–y 5Step 1 x 2y –1Solve the first equation for x bysubtracting 2y from both sides. 2y 2yx –2y – 1Step 2 x – y 5(–2y – 1) – y 5–3y – 1 5Holt McDougal Algebra 1Substitute –2y – 1 for x in thesecond equation.Simplify.
Solving Systems by SubstitutionExample 1C ContinuedStep 3 –3y – 1 5 1 1–3y 6Solve for y.Add 1 to both sides.–3y 6–3 –3y –2Step 4 x – y 5x – (–2) 5x 2 5–2 –2x 3Step 5 (3, –2)Divide both sides by –3.Holt McDougal Algebra 1Write one of the originalequations.Substitute –2 for y.Subtract 2 from both sides.Write the solution as anordered pair.
Solving Systems by SubstitutionCheck It Out! Example 1aSolve the system by substitution.y x 3y 2x 5Step 1 y x 3y 2x 5Both equations are solved for y.Step 2 y x 32x 5 x 3Substitute 2x 5 for y in the firstequation.Step 3 2x 5 x 3–x – 5 –x – 5x –2Solve for x. Subtract x and 5from both sides.Holt McDougal Algebra 1
Solving Systems by SubstitutionCheck It Out! Example 1a ContinuedSolve the system by substitution.Step 4y x 3y –2 3y 1Write one of the originalequations.Substitute –2 for x.Step 5(–2, 1)Write the solution as anordered pair.Holt McDougal Algebra 1
Solving Systems by SubstitutionCheck It Out! Example 1bSolve the system by substitution.x 2y – 4x 8y 16Step 1 x 2y – 4Step 2 x 8y 16(2y – 4) 8y 16The first equation is solved for x.Step 3 10y – 4 16 4 410y 2010y20 1010y 2Simplify. Then solve for y.Add 4 to both sides.Holt McDougal Algebra 1Substitute 2y – 4 for x in thesecond equation.Divide both sides by 10.
Solving Systems by SubstitutionCheck It Out! Example 1b ContinuedSolve the system by substitution.Step 4x 8y 16x 8(2) 16x 16 16– 16 –16x 0Step 5 (0, 2)Holt McDougal Algebra 1Write one of the originalequations.Substitute 2 for y.Simplify.Subtract 16 from bothsides.Write the solution asan ordered pair.
Solving Systems by SubstitutionCheck It Out! Example 1cSolve the system by substitution.2x y –4x y –7Step 1x y –7–y –yx –y – 7Step 2x –y – 72(–y – 7) y –42(–y – 7) y –4–2y – 14 y –4Holt McDougal Algebra 1Solve the second equation for xby subtracting y from eachside.Substitute –y – 7 for x in thefirst equation.Distribute 2.
Solving Systems by SubstitutionCheck It Out! Example 1c ContinuedSolve the system by substitution.Step 3 –2y – 14 y –4–y – 14 –4 14 14–yStep 4 10y –10x y –7x (–10) –7x – 10 – 7Holt McDougal Algebra 1Combine like terms.Add 14 to each side.Write one of the originalequations.Substitute –10 for y.
Solving Systems by SubstitutionCheck It Out! Example 1c ContinuedSolve the system by substitution.Step 5x – 10 –7 10 10Add 10 to both sides.x 3Step 6Holt McDougal Algebra 1(3, –10)Write the solution as anordered pair.
Solving Systems by SubstitutionSometimes you substitute anexpression for a variable that has acoefficient. When solving for thesecond variable in this situation, youcan use the Distributive Property.Holt McDougal Algebra 1
Solving Systems by SubstitutionCautionWhen you solve one equation for a variable, youmust substitute the value or expression into theother original equation, not the one that had justbeen solved.Holt McDougal Algebra 1
Solving Systems by SubstitutionExample 2: Using the Distributive PropertySolvey 6x 11by substitution.3x 2y –5Step 1 y 6x 11– 6x – 6xy –6x 11Solve the first equation for yby subtracting 6x from eachside.Step 23x 2y –53x 2(–6x 11) –5Substitute –6x 11 for y in thesecond equation.3x 2(–6x 11) –5Distribute 2 to the expressionin parentheses.Holt McDougal Algebra 1
Solving Systems by SubstitutionExample 2 ContinuedSolvey 6x 11by substitution.3x 2y –5Simplify. Solve for x.Step 3 3x 2(–6x) 2(11) –53x – 12x 22 –5–9x 22 –5– 22 –22 Subtract 22 from–9x –27both sides.–9x –27 Divide both sidesby –9.–9–9x 3Holt McDougal Algebra 1
Solving Systems by SubstitutionExample 2 ContinuedSolveStep 4y 6x 11by substitution.3x 2y –5y 6x 11y 6(3) 11y 18 11–18 –18Write one of the originalequations.Substitute 3 for x.Simplify.Subtract 18 from each side.y –7Step 5Holt McDougal Algebra 1(3, –7)Write the solution as anordered pair.
Solving Systems by SubstitutionCheck It Out! Example 2Solve–2x y 83x 2y 9by substitution.Step 1 –2x y 8 2x 2xy 2x 8Solve the first equation for yby adding 2x to each side.Step 23x 2y 93x 2(2x 8) 9Substitute 2x 8 for y in thesecond equation.3x 2(2x 8) 9Holt McDougal Algebra 1Distribute 2 to the expressionin parentheses.
Solving Systems by SubstitutionCheck It Out! Example 2 ContinuedSolve–2x y 83x 2y 9by substitution.Step 3 3x 2(2x) 2(8) 9Simplify. Solve for x.3x 4x 16 97x 16 9–16 –167x –77x –777x –1Holt McDougal Algebra 1Subtract 16 fromboth sides.Divide both sidesby 7.
Solving Systems by SubstitutionCheck It Out! Example 2 ContinuedSolveStep 4–2x y 83x 2y 9–2x y 8–2(–1) y 8y 2 8–2 –2yStep 5Holt McDougal Algebra 1by substitution.Write one of the originalequations.Substitute –1 for x.Simplify.Subtract 2 from each side. 6(–1, 6)Write the solution as anordered pair.
Solving Systems by SubstitutionExample 3: Consumer Economics ApplicationJenna is deciding between two cell-phoneplans. The first plan has a 50 sign-up fee andcosts 20 per month. The second plan has a 30 sign-up fee and costs 25 per month. Afterhow many months will the total costs be thesame? What will the costs be? If Jenna has tosign a one-year contract, which plan will becheaper? Explain.Write an equation for each option. Let t representthe total amount paid and m represent the numberof months.Holt McDougal Algebra 1
Solving Systems by SubstitutionExample 3 ContinuedTotalpaidissignup feepayment for eachplus amount month.Option 1t 50 20mOption 2t 30 25mStep 1 t 50 20mt 30 25mBoth equations are solvedfor t.Step 2 50 20m 30 25mSubstitute 50 20m for t inthe second equation.Holt McDougal Algebra 1
Solving Systems by SubstitutionExample 3 ContinuedStep 3 50 20m 30 25m–20m– 20m50 30 5m–30–3020 5m20 5m55m 4Solve for m. Subtract 20mfrom both sides.Subtract 30 from bothsides.Step 4 t 30 25mWrite one of the originalequations.Substitute 4 for m.Simplify.t 30 25(4)t 30 100t 130Holt McDougal Algebra 1Divide both sides by 5.
Solving Systems by SubstitutionExample 3 ContinuedStep 5(4, 130)Write the solution as anordered pair.In 4 months, the total cost for each option would bethe same 130.If Jenna has to sign a one-year contract,which plan will be cheaper? Explain.Option 1: t 50 20(12) 290Option 2: t 30 25(12) 330Jenna should choose the first plan because it costs 290 for the year and the second plan costs 330.Holt McDougal Algebra 1
Solving Systems by SubstitutionCheck It Out! Example 3One cable television provider has a 60 setupfee and charges 80 per month, and thesecond has a 160 equipment fee and charges 70 per month.a. In how many months will the cost be thesame? What will that cost be.Write an equation for each option. Let trepresent the total amount paid and mrepresent the number of months.Holt McDougal Algebra 1
Solving Systems by SubstitutionCheck It Out! Example 3 ContinuedTotalpaidisfeepayment for eachplus amount month.Option 1t 60 80mOption 2t 160 70mStep 1 t 60 80mt 160 70mBoth equations are solvedfor t.Step 2 60 80m 160 70m Substitute 60 80m for t inthe second equation.Holt McDougal Algebra 1
Solving Systems by SubstitutionCheck It Out! Example 3 ContinuedStep 3 60 80m 160 70m Solve for m. Subtract 70m–70m–70m from both sides.60 10m 160Subtract 60 from both–60–60sides.10m 100Divide both sides by 10.1010m 10Write one of the originalStep 4 t 160 70mequations.t 160 70(10)Substitute 10 for m.t 160 700Simplify.t 860Holt McDougal Algebra 1
Solving Systems by SubstitutionCheck It Out! Example 3 ContinuedStep 5 (10, 860) Write the solution as an ordered pair.In 10 months, the total cost for each optionwould be the same, 860.b. If you plan to move in 6 months, which isthe cheaper option? Explain.Option 1: t 60 80(6) 540Option 2: t 160 70(6) 580The first option is cheaper for the first six months.Holt McDougal Algebra 1
Solving Systems by SubstitutionLesson Quiz: Part ISolve each system by substitution.1.2.3.y 2x(–2, –4)x 6y – 113x – 2y –1–3x y –1x–y 4Holt McDougal Algebra 1(1, 2)
Solving Systems by SubstitutionLesson Quiz: Part II4. Plumber A charges 60 an hour. Plumber Bcharges 40 to visit your home plus 55 foreach hour. For how many hours will the totalcost for each plumber be the same? How muchwill that cost be? If a customer thinks they willneed a plumber for 5 hours, which plumbershould the customer hire? Explain.8 hours; 480; plumber A: plumber A ischeaper for less than 8 hours.Holt McDougal Algebra 1
Holt McDougal Algebra 1 Solving Systems by Substitution Solving Systems of Equations by Substitution Step 2 Step 3 Step 4 Step 5 Step 1 Solve for one variable in at least one equation, if necessary. Substitute the resulting expression into the other equation. So
