The Elimination Method

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Madison Lawrence
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1 The Elimination Method The elimination method is sometimes called the addition method. It involves the process of eliminating a variable through addition. It is the final method of solving a sstem that we will consider. This method is based on the notion that if a b and c d, the c b d. As an eample of this consider if and, the. Procedure: 1. Write both equations in General Form A B C. Choose the variable to eliminate. If necessar, multipl one or both equations b appropriate constants so that the coefficients of the variable to be eliminated are opposites.. Add the equations. Then solve the resulting equation for the remaining variable.. Substitute the value found in the previous step in either of the original equations and solve for the other variable. Eample: Solve the sstem 6 10 b the elimination method. 1. Both equations are alread in general form.. Because the variables have opposite coefficients, the should be eliminated.. Not necessar Solve for the remaining variable using either of the original equations. 6 The solution is the ordered pair -, 8). 6 8

2 Eample: Solve the sstem b the elimination method. 1. Both equations are alread in general form.. Because the variables have same coefficients, the should be eliminated.. Multipl the 1 st equation b -1). 1) ) ) 1) 0. Solve for the remaining variable using either of the original equations. ) The solution is the ordered pair,-). Eample: Solve the sstem 1 18 b the elimination method. 1. Both equations are alread in general form.. Because the coefficient of the variable in the 1 st equation is 1, the variable should be eliminated.. Multipl the 1 st equation b -) ) 18) ) )

3 . Solve for the remaining variable using either of the original equations. 18 6) 18 0 The solution is the ordered pair 0, 6). Eample: Solve the sstem 16 b the elimination method. 1. Both equations are alread in general form.. Eliminate the variable. Note: This choice of variable is completel arbitrar.. Multipl the 1 st equation b -) and the nd equation b ). 16 ) ) ) ) ) ) 16)) Solve for the remaining variable using either of the original equations. 16 ) 16 The solution is the ordered pair, -). Eample: Solve the sstem 9 1 b the elimination method. 1. Rewrite the 1st equation in general form

4 . Eliminate the variable.. Multipl the 1 st equation b -) 9 1 ) ) ) ) ) This is an identit. When the resulting equation is an identit this means that there are an infinite number of solutions dependent sstem.) Eample: Solve the sstem 0 6 b the elimination method. 1. Rewrite the nd equation in general form Eliminate the variable. Note: This choice of variable is completel arbitrar.. Multipl the 1 st equation b ) and the nd equation b ). 0 6 ) ) 0)) ) ) 6)) Solve for the remaining variable using either of the original equations. 6 8) The solution is the ordered pair 10, 8).

5 6 1 0 Eample: Solve the sstem b the elimination method. 1. Rewrite the 1st equation in general form Eliminate the variable.. Multipl the nd equation b -) ) ) ) ) This is a contradiction. When the resulting equation is a contradiction this means that there are no solutions inconsistent sstem.) Applications: Eample: The Phoeni Zoo has different admission prices for adults and children. When Mr. and Mrs. Coughlin went with their children, the bill was $.00. If Mrs. Weaver and her children got in for $18.0, then what is the price of an adults ticket and what is the price of a child s ticket? Let c the cost of a child s ticket and a the cost of an adult ticket. Then we can create the following equations. a c Solve for c b using elimination. a c 18.0 a c a c 18.0 a c ) a c) 18.0) ) a c a 6c c Solve for a using substitution. a c 18.0 a ) 18.0 a 6.0 Therefore, the cost of an adult ticket is $6.0 and a child s ticket is $.00.

6 Eample: A farmer has some cows and ostriches. One da he observed that his animals, which are normal, have 8 ees and 1 legs. How man animals of each tpe does he have? Let c the number of cows and o the number of ostriches. Then we can create the following equations. c o 8 Solve for c b using elimination. c o 1 c o 8 c o 1 )c o) 8) ) c o 1 c o 168 c o 1 o Solve for c. c o 8 c ) 8 c 8 c 19 There are 19 cows and ostriches. Eample: Mr. Thomas and his three children paid a total of $6. for admission to Wet and Wild Water park. Mr. and Mrs. Bennett and their si children paid a total of $11.0. What is the price of an adult ticket and what is the price of a child s ticket? Let a the price of an adult ticket and c the price of a child s ticket. Then we can create the following equations. a c 6. Solve this sstem b elimination. a 6c 11.0 a c 6. a 6c 11.0 ) a c) 6.) ) a 6c 11.0 a 6c 11.0 a 6c a 0c Since this is an identit, there are an infinite number of possible solutions.

7 Eample: At Grand Canon Universit, there are more males than females enrolled in MAT 0. Twothirds of the males and two-thirds of the females are graduating seniors. If there are 0 more graduating senior males than graduating senior females, how man males and how man females are there enrolled in MAT 0? Let m the number of male students and f the number of females students enrolled in MAT 0. We can then create the following equations. m m f f 0 m m f f 0 Multipl the nd equation b to eliminate the fractions. m f m f 0 m f m f 90 Multipl the 1 st equation b -). m f ) m f ) ) ) m f 90 m f 90 m f m f Add the two equations together. m f m f 0m 0 f Since this is a contradiction, there is no solution to the problem.

MA 15800, Summer 2016 Lesson 25 Notes Solving a System of Equations by substitution (or elimination) Matrices. 2 A System of Equations

MA 15800, Summer 2016 Lesson 25 Notes Solving a System of Equations by substitution (or elimination) Matrices. 2 A System of Equations MA 800, Summer 06 Lesson Notes Solving a Sstem of Equations b substitution (or elimination) Matrices Consider the graphs of the two equations below. A Sstem of Equations From our mathematics eperience,