7.4 Using a Substitution Strategy to Solve a System of Linear Equations
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1 7.4 Using a Substitution Strategy to Solve a System of Linear Equations FOCUS Use substitution to solve a linear system. One algebraic strategy is to solve by substitution. We use this strategy when one variable has coefficient 1. Example 1 Solving a Linear System by Substitution Solve this linear system. 2x y 3 1 4x 3y 5 2 We number the equations in a linear system to be able to refer to them easily. Solution 2x y 3 1 4x 3y 5 2 In equation 1, the coefficient of y is 1. So, solve equation 1 for y. 2x y 3 Subtract 2x from each side. y 3 2x Substitute y 3 2x in equation 2. 4x 3y 5 2 4x 3(3 2x) 5 Multiply to remove the brackets. 4x 9 6x 5 Combine like terms. 2x 9 5 Isolate 2x. Subtract 9 from each side. 2x 5 9 2x 4 Solve for x. Divide each side by 2. x 2 To find the value of y when x 2, substitute in one of the given equations. Choose equation 1. 2x y 3 1 Substitute: x 2 2(2) y 3 Solve for y. 4 y 3 y 3 4 y 1 The solution is: x 2 and y Copyright 2011 Pearson Canada Inc.
2 Verify the solution. In each equation, substitute: x 2 and y 1 2x y 3 L.S. 2x y R.S. 3 2(2) ( 1) For each equation, L.S. R.S. 4x 3y 5 L.S. 4x 3y 4(2) 3( 1) So, the solution of the linear system is x 2 and y 1. R.S. 5 Check 1. Solve this linear system. x 3y 2 1 3x 5y 6 2 In equation 1, the coefficient of x is 1. So, solve the equation for x. x 3y 2 x Substitute x in equation 2. 3x 5y 6 2 3( ) 5y 6 Multiply to remove the brackets. y To find the value of x when y, substitute in equation 1. x 3y 2 1 x 3( ) 2 x The solution is: x and y Verify the solution. In each equation, substitute: x and y x 3y 2 3x 5y 6 L.S. R.S. For each equation, L.S. R.S. L.S. So, the solution of the linear system is x and y. R.S Copyright 2011 Pearson Canada Inc. 401
3 Example 2 Using a Linear System to Solve a Problem a) Create a linear system to model this situation: Tickets are sold for the Senior Safari Day at the Greater Vancouver Zoo. Meryl buys 5 admission tickets and 3 train tickets. She pays $65. Howard buys 2 admission tickets and 1 train ticket. He pays $25. b) Solve this problem: What is the price of each type of ticket? Solution a) Cost of 5 admission tickets cost of 3 train tickets $65 Cost of 2 admission tickets cost of 1 train ticket $25 Let the cost of an admission ticket be a dollars and the cost of a train ticket be t dollars. A linear system that models the situation is: 5a 3t a t 25 2 b) Solve the linear system. In equation 2, the coefficient of t is 1. So, solve equation 2 for t. 2a t 25 2 t 25 2a Substitute t 25 2a in equation 1. 5a 3t a 3(25 2a) 65 5a 75 6a 65 a a a 10 a 10 Multiply to remove the brackets. Combine like terms. Isolate a. Subtract 75 from each side. Solve for a. Multiply each side by 1. To find the value of t when a 10, substitute in equation 2. 2a t (10) t t 25 t 5 The solution is: a 10 and t 5 Use the data in the problem to verify these numbers. Cost of 5 admission tickets at $10 each and 3 train tickets at $5 each: $50 $15 $65 Cost of 2 admission tickets at $10 each and 1 train ticket at $5: $20 $5 $25 The total costs match the data in the problem. So, the solution is correct. An admission ticket costs $10 and a train ticket costs $5. It s a good idea when checking the solution to a problem to use the original problem, not the equations Copyright 2011 Pearson Canada Inc.
4 Check 1. a) Create a linear system to model this situation: A math test has short-answer questions and word problems. A short-answer question is worth 2 marks and a word problem is worth 4 marks. There are 11 questions for a total of 30 marks. Let the number of short-answer questions be s, and the number of word problems be w. Short answer Word problem Total Marks per question Number of questions Number of marks The number of questions is represented by equation 1: The total number of marks is represented by equation 2: A linear system that models the situation is: 1 2 b) Solve this problem: How many short-answer questions and how many word problems are on the test? In equation 1, the coefficients of both s and w are. Solve equation 1 for w. w Substitute w in equation 2. 2s 4w s 4( ) 30 Since the coefficient of s is also 1, we could have solved for s. s To find the value of w when s, substitute in equation 1. w The solution is: s and w Use the data in the problem to verify the solution. short-answer questions worth marks each marks word problems worth marks each marks questions for a total of This matches the data in the problem, so the solution is correct. marks There are short-answer questions and word problems on the test Copyright 2011 Pearson Canada Inc. 403
5 Practice 1. For each linear system, the value of one variable in the solution is given. Find the value of the other variable. a) x 2y 1 1 b) 3x y x 3y x y 8 2 Given: y 1 Given: x 5 Substitute y in equation 1. Substitute x in equation 2. x 2y 1 x 2( ) 1 2x y 8 x y 2. Solve this linear system. y x 6 1 3x 2y 13 2 Substitute y x 6 in equation 2. 3x 2y x 2( ) 13 Verify the solution. In each equation, substitute: x and y y x 6 3x 2y 13 L.S. x To find the value of y when x, substitute in equation 1. y x 6 1 y 6 y The solution is: x and y R.S. For each equation, L.S. R.S. L.S. So, the solution of the linear system is x and y. R.S Copyright 2011 Pearson Canada Inc.
6 3. Solve this linear system. 3x 2y 25 1 x 2y 5 2 In equation 2, the coefficient of x is 1. So, solve equation 2 for x. x 2y 5 2 x _ Substitute x _ in equation 1. 3x 2y ( ) 2y 25 y To find the value of x when y, substitute in equation 2. x 2y 5 2 x 2( ) 5 x The solution is: x and y Verify the solution. In each equation, substitute: x and y 3x 2y 25 x 2y 5 L.S. R.S. L.S. R.S. L.S. L.S. For each equation, L.S. R.S. So, the solution of the linear system is x and y Copyright 2011 Pearson Canada Inc. 405
7 4. a) Create a linear system to model this situation: Michelle and Marty spend the afternoon at the local fair. Michelle rides the roller coaster 3 times and the super swing 5 times. She pays $25. Marty rides the roller coaster 5 times and the super swing once. He pays $27. Cost of roller coasters cost of super swings Cost of roller coasters cost of super swing Let the cost of a roller coaster ride be dollars. Let the cost of a super swing ride be dollars. A linear system that models the situation is: 1 2 b) Solve this problem: What is the cost of each type of ride? The solution is: and Use the data in the problem to verify the solution. A roller coaster ride costs and a super swing ride costs Copyright 2011 Pearson Canada Inc.
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