3-4 Study Guide and Intervention Systems of Equations in Three Variables Systems in Three Variables Use the methods used for solving systems of linear equations in two variables to solve systems of equations in three variables. A system of three equations in three variables can have a unique solution, infinitely many solutions, or no solution. A solution is an ordered triple. Example: Solve the system of equations. 3x + = y z – –6 2x + 2z = 8 y – 4x + 3z = y – –21 Step 1 Use elimination to make a system of two equations in two variables. 3x + = 2x + 2z = 8 y z y – –6 – (+) 2x + 2z = 8 (+) 4x + 3z = y y – – –21 Step 2 Solve the system of two equations. 5x + = 2 z (+) 6x = z – –13 Step 3 Substitute Examples Solve each system of equations. 1. 2x + 3y = 0 2. 2x + 4z = 11 3. 2y + = 8 z y x z – – – 2y 4z = 14 + 2y 6z = 2x + = 0 x x y z – – – –11 – 3x + 8z = 17 3x 2y = 11 3x 6y + 3z = 24 y – – –10z –
3-4 Study Guide and Intervention Systems of Equations in Three Variables Systems in Three Variables Use the methods used for solving systems of linear equations in two variables to solve systems of equations in three variables. A system of three equations in three variables can have a unique solution, infinitely many solutions, or no solution. A solution is an ordered triple. Example: Solve the system of equations. 3x + y – z = –6 2x – y + 2z = 8 4x + y – 3z = –21 Step 1 Use elimination to make a system of two equations in two variables. 3x + y – z = –6 2x – y + 2z = 8 (+) 2x – y + 2z = 8 (+) 4x + y – 3z = –21 Step 2 Solve the system of two equations. 5x + z = 2 (+) 6x – z = –13 Step 3 Substitute Examples Solve each system of equations. 1. 2x + 3y – z = 0 2. 2x – y + 4z = 11 3. x – 2y + z = 8 x – 2y – 4z = 14 x + 2y – 6z = –11 2x + y – z = 0 3x + y – 8z = 17 3x – 2y –10z = 11 3x – 6y + 3z = 24
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