Systems of Linear Equations: Solving by Substitution


 Frederick McKenzie
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1 8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing and addition as methods of solving linear sstems. A third method is called solution b substitution. Eample 1 Solving a Sstem b Substitution Solve b substitution. 12 (1) 3 (2) Notice that equation (2) sas that and 3 name the same quantit. So we ma substitute 3 for in equation (1). We then have Replace with 3 in equation (1). NOTE The resulting equation contains onl the variable, so substitution is just another wa of eliminating one of the variables from our sstem. NOTE The solution for a sstem is written as an ordered pair We can now substitute 3 for in equation (1) to find the corresponding coordinate of the solution So (3, 9) is the solution. This last step is identical to the one ou saw in Section 8.2. As before, ou can substitute the known coordinate value back into either of the original equations to find the value of the remaining variable. The check is also identical. CHECK YOURSELF 1 Solve b substitution. 9 4 The same technique can be readil used an time one of the equations is alread solved for or for, as Eample 2 illustrates. 657
2 658 CHAPTER 8 SYSTEMS OF LINEAR EQUATIONS Eample 2 Solving a Sstem b Substitution Solve b substitution (1) 2 7 (2) Because equation (2) tells us that is 2 7, we can replace with 2 7 in equation (1). This gives NOTE Now is eliminated from the equation, and we can proceed to solve for. 2 3(2 7) We now know that 3 is the coordinate for the solution. So substituting 3 for in equation (2), we have And (3, 1) is the solution. Once again ou should verif this result b letting 3 and 1 in the original sstem. CHECK YOURSELF 2 Solve b substitution As we have seen, the substitution method works ver well when one of the given equations is alread solved for or for. It is also useful if ou can readil solve for or for in one of the equations. Eample 3 Solving a Sstem b Substitution Solve b substitution. 2 5 (1) 3 8 (2)
3 SYSTEMS OF LINEAR EQUATIONS: SOLVING BY SUBSTITUTION SECTION Neither equation is solved for a variable. That is easil handled in this case. Solving for in equation (1), we have NOTE Equation (2) could have been solved for with the result substituted into equation (1). 2 5 Now substitute 2 5 for in equation (2). 3(2 5) Substituting 1 for in equation (2) ields 3 ( 1) So (3, 1) is the solution. You should check this result b substituting 3 for and 1 for in the equations of the original sstem. CHECK YOURSELF 3 Solve b substitution Inconsistent sstems and dependent sstems will show up in a fashion similar to that which we saw in Section 8.2. Eample 4 illustrates this approach. Eample 4 Solving an Inconsistent or Dependent Sstem Solve the following sstems b substitution. (a) (1) 2 3 (2) NOTE Don t forget to change both signs in the parentheses. From equation (2) we can substitute 2 3 for in equation (1). 4 2(2 3) Both variables have been eliminated, and we have the true statement 6 6.
4 660 CHAPTER 8 SYSTEMS OF LINEAR EQUATIONS Recall from the last section that a true statement tells us that the lines coincide. We call this sstem dependent. There are an infinite number of solutions. (b) (3) 2 2 (4) Substitute 2 2 for in equation (3). 3(2 2) This time we have a false statement. This means that the sstem is inconsistent and that the graphs of the two equations are parallel lines. There is no solution. CHECK YOURSELF 4 Indicate whether the sstems are inconsistent (no solution) or dependent (an infinite number of solutions). (a) (b) The following summarizes our work in this section. Step b Step: To Solve a Sstem of Linear Equations b Substitution Step 1 Step 2 Step 3 Step 4 Step 5 Solve one of the given equations for or. If this is alread done, go on to step 2. Substitute this epression for or for into the other equation. Solve the resulting equation for the remaining variable. Substitute the known value into either of the original equations to find the value of the second variable. Check our solution in both of the original equations. You have now seen three different was to solve sstems of linear equations: b graphing, adding, and substitution. The natural question is, Which method should I use in a given situation? Graphing is the least eact of the methods, and solutions ma have to be estimated. The algebraic methods addition and substitution give eact solutions, and both will work for an sstem of linear equations. In fact, ou ma have noticed that several eamples in this section could just as easil have been solved b adding (Eample 3, for instance). The choice of which algebraic method (substitution or addition) to use is ours and depends largel on the given sstem. Here are some guidelines designed to help ou choose an appropriate method for solving a linear sstem.
5 SYSTEMS OF LINEAR EQUATIONS: SOLVING BY SUBSTITUTION SECTION Rules and Properties: Choosing an Appropriate Method for Solving a Sstem 1. If one of the equations is alread solved for (or for ), then substitution is the preferred method. 2. If the coefficients of (or of ) are the same, or opposites, in the two equations, then addition is the preferred method. 3. If solving for (or for ) in either of the given equations will result in fractional coefficients, then addition is the preferred method. Eample 5 Choosing an Appropriate Method for Solving a Sstem Select the most appropriate method for solving each of the following sstems. (a) Addition is the most appropriate method because solving for a variable will result in fractional coefficients. (b) Substitution is the most appropriate method because the second equation is alread solved for. (c) Addition is the most appropriate method because the coefficients of are opposites. CHECK YOURSELF 5 Select the most appropriate method for solving each of the following sstems. (a) (b) (c) (d) Number problems, such as those presented in Chapter 2, are sometimes more easil solved b the methods presented in this section. Eample 6 illustrates this approach. Eample 6 Solving a Number Problem b Substitution The sum of two numbers is 25. If the second number is 5 less than twice the first number, what are the two numbers?
6 662 CHAPTER 8 SYSTEMS OF LINEAR EQUATIONS NOTE 1. What do ou want to find? 2. Assign variables. This time we use two letters, and. 3. Write equations for the solution. Here two equations are needed because we have introduced two variables. Step 1 Step 2 Step 3 25 The sum is You want to find the two unknown numbers. Let the first number and the second number. The second number is 5 less than twice the first. 4. Solve the sstem of equations. NOTE We use the substitution method because equation (2) is alread solved for. 5. Check the result. Step 4 25 (1) 2 5 (2) Substitute 2 5 for in equation (1). (2 5) From equation (1), The two numbers are 10 and 15. Step 5 The sum of the numbers is 25. The second number, 15, is 5 less than twice the first number, 10. The solution checks. CHECK YOURSELF 6 The sum of two numbers is 28. The second number is 4 more than twice the first number. What are the numbers? Sketches are alwas helpful in solving applications from geometr. Let s look at such an eample. Eample 7 Solving an Application from Geometr The length of a rectangle is 3 meters (m) more than twice its width. If the perimeter of the rectangle is 42 m, find the dimensions of the rectangle. Step 1 You want to find the dimensions (length and width) of the rectangle.
7 SYSTEMS OF LINEAR EQUATIONS: SOLVING BY SUBSTITUTION SECTION NOTE We used and as our two variables in the previous eamples. Use whatever letters ou want. The process is the same, and sometimes it helps ou remember what letter stands for what. Here L length and W width. Step 2 problem. W Let L be the length of the rectangle and W the width. Now draw a sketch of the L W Step 3 L Write the equations for the solution. L 2W 3 3 more than twice the width 2L 2W 42 The perimeter Step 4 Solve the sstem. NOTE Substitution is used because one equation is alread solved for a variable. L 2W 3 (1) 2L 2W 42 (2) From equation (1) we can substitute 2W 3 for L in equation (2). 2(2W 3) 2W 42 4W 6 2W 42 6W 36 W 6 Replace W with 6 in equation (1) to find L. L The length is 15 m, the width is 6 m. 6 m 15 m Step 5 Check these results. The perimeter is 2L 2W, which should give us 42 m. 2(15) 2(6) CHECK YOURSELF 7 The length of each of the two equal legs of an isosceles triangle is 5 in. less than the length of the base. If the perimeter of the triangle is 50 in., find the lengths of the legs and the base.
8 664 CHAPTER 8 SYSTEMS OF LINEAR EQUATIONS CHECK YOURSELF ANSWERS 1. ( 3, 12) 2. (6, 2) 3. (2, 1) 4. (a) Inconsistent sstem; (b) Dependent sstem 5. (a) Addition; (b) Substitution; (c) Substitution; (d) Addition 6. The numbers are 8 and The legs have length 15 in.; the base is 20 in.
9 Name 8.3 Eercises Section Date Solve each of the following sstems b substitution ANSWERS
10 ANSWERS Solve each of the following sstems b using either addition or substitution. If a unique solution does not eist, state whether the sstem is dependent or inconsistent Solve each sstem
11 ANSWERS Solve each of the following problems. Be sure to show the equation used for the solution. 47. Number problem. The sum of two numbers is 100. The second is three times the first. Find the two numbers Number problem. The sum of two numbers is 70. The second is 10 more than 3 times the first. Find the numbers. 49. Number problem. The sum of two numbers is 56. The second is 4 less than twice the first. What are the two numbers? 50. Number problem. The difference of two numbers is 4. The larger is 8 less than twice the smaller. What are the two numbers? Number problem. The difference of two numbers is 22. The larger is 2 more than 3 times the smaller. Find the two numbers. 52. Number problem. One number is 18 more than another, and the sum of the smaller number and twice the larger number is 45. Find the two numbers. 53. Number problem. One number is 5 times another. The larger number is 9 more than twice the smaller. Find the two numbers. 54. Package weight. Two packages together weigh 32 kilograms (kg). The smaller package weighs 6 kg less than the larger. How much does each package weigh? 55. Appliance costs. A washerdrer combination costs $1200. If the washer costs $220 more than the drer, what does each appliance cost separatel? 56. Voting trends. In a town election, the winning candidate had 220 more votes than the loser. If 810 votes were cast in all, how man votes did each candidate receive? 667
12 ANSWERS Cost of furniture. An office desk and chair together cost $850. If the desk cost $50 less than twice as much as the chair, what did each cost? a. b. 58. Dimensions of a rectangle. The length of a rectangle is 2 inches (in.) more than twice its width. If the perimeter of the rectangle is 34 in., find the dimensions of the rectangle. 59. Perimeter. The perimeter of an isosceles triangle is 37 in. The lengths of the two equal legs are 6 in. less than 3 times the length of the base. Find the lengths of the three sides. 60. You have a parttime job writing the Consumer Concerns column for our local newspaper. Your topic for this week is clothes drers, and ou are planning to compare the Helpmate and the Whirlgarb drers, both readil available in stores in our area. The information ou have is that the Helpmate drer is listed at $520, and it costs 22.5 to dr an average size load at the utilit rates in our cit. The Whirlgarb drer is listed at $735, and it costs 15.8 to run for each normal load. The maintenance costs for both drers are about the same. Working with a partner, write a short article giving our readers helpful advice about these appliances. What should the consider when buing one of these clothes drers? Getting Read for Section 8.4 [Section 2.7] Graph the solution sets for the following linear inequalities. (a) 8 (b)
13 ANSWERS (c) (d) 2 c. d. e. f. (e) 3 (f) 5 Answers 1. (2, 8) 3. (4, 2) (4, 1) 9. (10, 6) 11. (5, 3) 13. (10, 2) No solution 19. (3, 3) (4, 1) 25. Infinite number of solutions 27. (10, 1) 29. ( 2, 1) 31. (0, 2) 33. (2, 3) 35. Dependent sstem , 3 1 3, 2 3 2, 3 4 3, (0, 10) 45. (2, 1) , , , , Washer $710, drer $ Desk $550, chair $ in., 15 in., 15 in. a. 8 b , , 2 669
14 c d. 2 e. 3 f
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