CHAPTER THREE: SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES ALGEBRA TWO Section Solving Systems of Linear Equations in Three Variables

LEARNING GOALS Goal One - Solve systems of linear equations in three variables. Goal Two - Use linear systems in three variables to model real- life situations.

VOCABULARY  A system of three linear equations includes three equations in the same variables.  A solution of a linear system in three variables is an ordered triple (x, y, z) that satisfies all three equations. The linear combination method you learned in Lesson 3.2 can be extended to solve a system of linear equations in three variables.

ALGEBRAIC SOLUTIONS THE LINEAR COMBINATION METHOD (THREE VARIABLE SYSTEMS) (THREE VARIABLE SYSTEMS) STEP 1: Use the linear combination method to rewrite the linear system in three variables as a linear system in two variables. STEP 2: Solve the new linear system for both of its variables. STEP 3: Substitute the values found in Step 2 into one of the original equations and solve for the third variable.

ALGEBRAIC SOLUTIONS THE LINEAR COMBINATION METHOD (THREE VARIABLE SYSTEMS) (THREE VARIABLE SYSTEMS) NOTE: If you obtain a false equation, such as 0 = 1, in any step, then the system has no solution. If you do not obtain a false solution, but obtain an identity, such as 0 = 0, then the system has infinitely many solutions.

Using a Linear Combination Method Solve the system: x + y + z = 2 Equation 1 -x + 3y + 2z = 8 Equation 2 4x + y = 4Equation 3 Since Equation 3 does not have a z-term, eliminate the z from one of the other equations. x + y + z = 2 -x + 3y + 2z = 8 times -2 times 1 -2x - 2y - 2z = -4 -x + 3y + 2z = 8 -3x + y = 4 Add the equations.

Using a Linear Combination Method Solve the system: x + y + z = 2 Equation 1 -x + 3y + 2z = 8 Equation 2 4x + y = 4Equation 3 Now use this new equation with Equation 3 to solve for x and y. -3x + y = 4 4x + y = 4 times -1 times 1 3x - y = -4 4x + y = 4 7x = 0 Add the equations. x = 0

Using a Linear Combination Method Solve the system: x + y + z = 2 Equation 1 -x + 3y + 2z = 8 Equation 2 4x + y = 4Equation 3 Substitute x = 0 and solve for y. 0 -3(0) + y = 4 y = 4

Using a Linear Combination Method Solve the system: x + y + z = 2 Equation 1 -x + 3y + 2z = 8 Equation 2 4x + y = 4Equation 3 Substitute x = 0, y = 4 and solve for z. x + y + z = 2 z = (0) + (4) + z = 2 The solution is the ordered triple (0, 4, -2).

ALGEBRAIC SOLUTIONS THE LINEAR COMBINATION METHOD (THREE VARIABLE SYSTEMS) NOTE: This technique requires careful, tedious attention to the detail of the steps involved. The best way to master the technique is to PRACTICE, PRACTICE, PRACTICE!!!!!!!!!

Solve the system: 3x+2y+4z = 11equation 1. 2x-y+3z = 4equation 2. 5x-3y+5z= -1equation 3.

Solve the system: 3x+2y+4z = 11equation 1. 2x-y+3z = 4equation 2. 5x-3y+5z= -1equation 3.

ASSIGNMENT READ & STUDY: pg WRITE: pg #13, #17, #19, #23, #25, #29, #31, #33