ALGEBRA TWO CHAPTER THREE SYSTEMS OF LINEAR EQUATIONS

ALGEBRA TWO CHAPTER THREE: SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES Section 3. 6 - 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 reallife 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) 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) 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 -x + 3 y + 2 z = 8 4 x + y = 4 Equation 1 Equation 2 Equation 3 Since Equation 3 does not have a z-term, eliminate the z from one of the other equations. x + y + z = 2 times -2 -x + 3 y + 2 z = 8 times 1 Add the equations. -2 x - 2 y - 2 z = -4 -x + 3 y + 2 z = 8 -3 x + y = 4

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

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

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

ALGEBRAIC SOLUTIONS THE LINEAR COMBINATION METHOD VARIABLE SYSTEMS) (THREE 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!!!!!

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

ASSIGNMENT READ & STUDY: pg. 177 -180. WRITE: pg. 181 -184. #13, #17, #19, #23, #25, #29, #31, #33