SECTION 11.1 Systems of Linear Equations: Substitution and Elimination 759 In Words When using elimination, get the coefficient of one variable to be opposite that of the other. Rules for Obtaining an Equivalent System of Equations • Interchange any two equations of the system. • Multiply (or divide) both sides of an equation by the same nonzero constant. • Replace any equation in the system by the sum (or difference) of that equation and a nonzero multiple of any other equation in the system. The idea behind the method of elimination is to replace the original system of equations by an equivalent system so that adding two of the equations eliminates a variable. The rules for obtaining equivalent equations are the same as those studied earlier.We may also interchange any two equations of the system and/or replace any equation in the system by the sum (or difference) of that equation and a nonzero multiple of any other equation in the system. An example will give you the idea. As you work through the example, pay particular attention to the pattern being followed. How to Solve a System of Linear Equations by Elimination Solve: + = − + =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 3 1 3 (1) (2) EXAMPLE 5 Step-by-Step Solution Step 1 Multiply both sides of one or both equations by a nonzero constant so that the coefficients of one of the variables are additive inverses. Multiply equation (2) by 2 so that the coefficients of x in the two equations are additive inverses. + = − + =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 3 1 3 ( ) ( ) + = − + = − ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 3 1 2 2 3 + = − + =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 3 1 2 2 6 Step 2 Add the equations to eliminate the variable. Solve the resulting equation. + = − + =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 3 1 2 2 6 =− =− y y 5 5 1 Step 3 Back-substitute the value of the variable found in Step 2 into one of the original equations to find the value of the remaining variable. Back-substitute = − y 1 into equation (1) and solve for x. ( ) + = + − = − = = = x y x x x x 2 3 1 2 3 1 1 2 3 1 2 4 2 Step 4 Check the solution found. The check is left to you. The solution of the system is = x 2 and = − y 1. The solution also can be written as the ordered pair ( ) − 2, 1. (1) (2) (1) (2) Multiply by 2. (1) (2) (1) (2) Add equations (1) and (2). Solve for y. Equation (1) Substitute y 1. =− Simplify. Add 3 to both sides. Solve for x. Now Use Elimination to Work PROBLEM 21
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