6.7 Solving Systems of Linear Equations 357 to both equations and so there are an infinite number of solutions to the system. This agrees with the answer we obtained algebraically in Example 5. When solving a system of equations, if you obtain a true statement, such as 0 0 = or 6 6, = the system is dependent and has an infinite number of solutions. Learning Catalytics Keyword: Angel-SOM-6.7 (See Preface for additional details.) Addition Method If neither of the equations in a system of linear equations has a variable with a coefficient of 1, it is generally easier to solve the system by using the addition (or elimination) method. To solve a system of linear equations by the addition method, it is necessary to obtain two equations whose sum will be a single equation containing only one variable. To achieve this goal, we rewrite the system of equations as two equations where the coefficients of one of the variables are opposites of each other. For example, if one equation has a term of x2 , we might rewrite the other equation so that its x term will be x2 . − To obtain the desired equations, it might be necessary to multiply one or both equations in the original system by a number. When an equation is to be multiplied by a number, we will place brackets around the equation and place the number the equation is to be multiplied by before the brackets. For example, x y 4[2 3 6] + = means that each term on both sides of the equal sign in the equation x y 2 3 6 + = is to be multiplied by 4: + = + = x y x y 4[2 3 6] gives 8 12 24 This notation will make our explanations much more efficient and easier for you to follow. SOLVING A SYSTEM OF EQUATIONS BY THE ADDITION METHOD 1. If necessary, rewrite the equations so that the terms containing the variables appear on one side of the equal sign and the constants appear on the other side of the equal sign. 2. If necessary, multiply one or both equations by a constant(s) so that when you add the equations, the sum will be an equation containing only one variable. 3. Add the equations to obtain a single equation in one variable. 4. Solve for the variable in the equation obtained in Step 3. 5. Substitute the value determined in Step 4 into either of the original equations and solve for the other variable. PROCEDURE Example 6 Eliminating a Variable by the Addition Method Solve the following system of equations by the addition method. x y x y 4 13 2 5 + = − = Solution Since the coefficients of the y-terms, 1 and 1, − are opposites, the sum of the y-terms will be zero when the equations are added. Thus, the sum of the two equations will contain only one variable, x. Add the two equations to obtain one equation in one variable. Then solve for the remaining variable. x y x y x x 4 13 2 5 6 18 3 + = − = = =
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