Solving Linear Systems of Equations Three Variables Recall

Solving Linear Systems of Equations - Three Variables • Recall that to solve the linear system of equations in two variables. . . we needed to find the values of x and y that satisfied both equations. • Now consider a linear system of equations in three variables. . . • The goal of solving this system is to find the values of x, y, and z that satisfy all three equations. Table of Contents

Solving Linear Systems of Equations - Three Variables • Example 1: Solve the following system of equations. • First number the equations. . . • Then select any two of the equations and use the addition method to eliminate one variable, say # 1 and # 2 to eliminate z. . . Table of Contents Slide 2

Solving Linear Systems of Equations - Three Variables • Now use a different pair of equations and eliminate the same variable z. • Multiply equation # 2 by 2 and add the result to equation # 3. . . Table of Contents Slide 3

Solving Linear Systems of Equations - Three Variables • The two equations that resulted from the previous work are. . . • This is simply a system of two linear equations in two unknowns, which is easily solved. . . Table of Contents Slide 4

Solving Linear Systems of Equations - Three Variables • Now substitute the solutions found for x and y in any one of the three original equations, say # 1, to solve for z. . . • Written as an ordered triple, ( 1, 2, -1) is the solution to the system of equations. To check, the values could be entered into equations # 2 and # 3. Table of Contents Slide 5

Solving Linear Systems of Equations - Three Variables • Example 2: Solve the following system of equations. • First number the equations. . . • Eliminate the variable y by adding equations # 1 and # 2. . . Table of Contents Slide 6

Solving Linear Systems of Equations - Three Variables • Now use equations # 1 and # 3 to eliminate y again. . . Table of Contents Slide 7

Solving Linear Systems of Equations - Three Variables • The two equations that resulted from the previous work are. . . • Solve this system. . . Table of Contents Slide 8

Solving Linear Systems of Equations - Three Variables • Substituting these values into equation # 1 yields. . . • The solution can be written as. . . Table of Contents Slide 9

Solving Linear Systems of Equations - Three Variables • The process of solving a system of three linear equations in three variables is outlined as follows: 1) Use the addition method on two of the three equations to eliminate one of the variables. 2) Use the addition method on another two equations (different from the pair used in step one) and eliminate the same variable that was eliminated in step one. Table of Contents Slide 10

Solving Linear Systems of Equations - Three Variables 3) The equations found in steps one and two form a system of two linear equations in two unknowns, which can be solved by previous methods. 4) Substitute the values found in step three into one of the three original equations to find the value of the third variable. Table of Contents Slide 11

Solving Linear Systems of Equations - Three Variables Table of Contents