Solving a System of Equations in Two Variables by the Addition Method

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Solving a System of Equations in Two Variables by the Addition Method Section 4.3 Solving a System of Equations in Two Variables by the Addition Method

Example Solve by addition. – x + y = 2 x + y = 4 – x + y = 2 x + y = 4 Add the two equations. y = 3 Divide each side by 2. x + y = 4 Substitute y = 3 into either equation. x + (3) = 4 x = 1 The solution is (1, 3). 2

The Addition Method Procedure for Solving a System of Equations by the Addition Method Multiply each term of one or both equations by some nonzero integer so that the coefficients of one of the variables are opposites. Add the equations of this new system so that one variable is eliminated. Solve the resulting equation for the remaining variable. Substitute this value found in step 3 into one of the original or equivalent equations to find the value of the other variable. Check the solution in both of the original equations. 3

Example Solve by addition. 5x – 3y = 14 2x – y = 6 Multiply each term of equation (2) by 3.  6x + 3y = 18 This equation is equivalent to equation (2). 5x – 3y = 14  6x + 3y = 18  x =  4 Add the two equations. Substitute this value into either equation to find y. x = 4 Continued 4

Example (cont) 5x – 3y = 14 2x – y = 6 2x – y = 6 2(4) – y = 6 Substitute. 8  y = 6 y = 2 The solution is (4, 2). y = 2 Be sure to check the solution in both original equations. 5

Example Solve by addition. Multiply each term of equation (1) by 8. This equation is equivalent to equation (1). Multiply each term of this new equation (1) by 3. Now the equations are ready to be added. Continued 6

Example (cont) Add the two equations. Substitute x = 1 into one of the equations. The solution is (1, 4). Be sure to check it. 7

Example Solve. Multiply each equation by 100. To eliminate the variable y, we want the coefficients of the y terms to be opposites. Continued 8

Example (cont) The solution is (–1, 2) and it checks. Multiply first equation by 3. Multiply the second equation by 5. Solve for x. Replace x by –1 in equations (3), and solve for y. The solution is (–1, 2) and it checks. 9