3.6 Solving Systems of Linear Equations in Three Variables

3.6 Solving Systems of Linear Equations in Three Variables

7.1 Solving Systems of Linear Equations in Three Variables Warm-Up No Solution Infinitely many solutions Here is a system of three linear equations in three variables: x 2 y 3z 3 2 x 5 y 4 z 13 5 x 4 y z 5 The ordered triple (2,-1,1) is a solution to this system since it is a

solution to all three equations. 2 2( 1) 3(1) 2 2 3 3 2(2) 5( 1) 4(1) 4 5 4 13 5(2) 4( 1) 1 10 4 1 5 The graph of a linear equation in three variables is a plane. Three planes in space can intersect in different ways (pg 218). The planes could The planes could intersect in a line. intersect in a The system has single point. infinitely many

The system has solutions exactly one solution The planes could have NO point of intersection. The left figure shows planes that intersect pairwise, but all 3 do not have a common point of intersection. The right figure shows parallel planes. Each system has NO solution. The linear combination method in lesson 3.2 can be extended to solve a system of linear equations in three variables. Solve this system 3x 2 y 4 z 11 2 x y 3 z 4 5 x 3 y 5 z 1

Our strategy will be to use two of the equations to eliminate one of the variables. We will then use two other equations to eliminate the same variable. Once we have two equations with two variables, we can use the technique we learned in lesson 3.2 Solve this system 3x 2 y 4 z 11 2 x y 3 z 4 5 x 3 y 5 z 1

Equation 1 Equation 2 Equation 3 Multiply Eq. 2 by 2 and add it to Eq. 1. Save this result 3 x 2 y 4 z 11 4 x 2 y 6 z 8 7x +10z = 19 Now multiply Eq. 2 by -3 and add it to Eq. 3. Save this result. 6 x 3 y 9 z 12

...5 x 3 y 5 z 1 -x -4z = -13 Solve this new system of linear equation in two variables. Multiply the bottom eq. by 7 and add it to the top eq. 7 x 10 z 19 7 x 28 z 91 -18z=-72 or z = 4 Substituting z=4 into either of the new equations will give x = -3finally substituting these values into any of the original equations give y = 2. Our final solution is (-3,2,4)

Here is a system with No Solution. Watch what happens when we try to solve it. x y z 2 3 x 3 y 3 z 14 x 2 y z 4 3 x 3 y 3 z 6 ..3 x 3 y 3 z 14 0=8 Equation 1 Equation 2 Equation 3 Add -3 times Eq 1 to Eq 2. Since this is a false equation, you can conclude the original system of equations

has no solution. Here is a system with MANY solutions. Watch what happens when we try to solve it. Equation 1 x y z 2 Equation 2 x y z 2 2 x 2 y z 4 Equation 3 x y z 2 x y z 2 2x + 2y =4

Add Eq. 1 to Eq. 2 New EQ. 1 x y z 2 Add Eq 2 to Eq 3 2 x 2 y z 4 3x +3y =6 New EQ 2 Solving this new system of two equation by adding -3 times the first eq. to 2 times the second eq. produces the identity 0 = 0. So, the system has infinitely many solution. You could describe the solution this way: divide New Eq 1 by 2 to get x+y=2, or

y=-x+2. Substituting this into the orignial Equation 1 produces z = 0. So any ordered triple of the form (x, -x+2,0) is a solution. For example (0,2,0) and (2,0,0) are solutions. Substitution Method x y z 24 5 x 3 y z 56 x y z Since x+y=z, substitute this for z in the first two equations x y ( x y ) 24 5 x 3 y ( x y ) 56 Simplify

2 x 2 y 24 6 x 4 y 56 Finally, solve this linear system of two equations and two variables to get x = 4 and y =8 Since z=x+y, z = 12. Our final solution is (4,8,12) www.pleasanton.k12.ca.us

Recently Viewed Presentations

  • We will start momentarily Essential Functions January 22,

    We will start momentarily Essential Functions January 22,

    Once essential functions are identified, a BPA identifies . how. all of the essential functions will be accomplished. Look at BPA process from point of view of both the overall process flow and the operational details. Focuses on clearly describing...
  • Scenarios & Transition Strategies - STEPS

    Scenarios & Transition Strategies - STEPS

    What assumptions are being made and which ones matter most? What metrics of change over time are required to assess the comparative likelihood of alternative energy pathways, including one dominated by shale oil and gas, meeting sustainability goals and timelines?
  • Promoting Disability Pride: Exploring the Evolution of ...

    Promoting Disability Pride: Exploring the Evolution of ...

    Promoting Disability Pride: Exploring the Evolution of Disability Culture IL Conversation. March 5, 2015. Facilitated by Steven E. Brown, Co-Founder, Institute on Disability Culture. Professor (Retired, currently Affiliate Faculty)
  • Alcohol and Other Drug (AODA) Task Force

    Alcohol and Other Drug (AODA) Task Force

    AODA Prevention Partnership May 8, 2013 University of Wisconsin-Stevens Point Agenda Welcome from your Co-Chairs Responsible Action legislation by SGA Alcohol-Wise Findings Preliminary AODA Survey Findings Online Program Options Sanction Grid Strategic Planning Process Next full Prevention Partnership meeting, November...
  • Environment and Health Sciences - United Nations

    Environment and Health Sciences - United Nations

    For example, the IASC Guidelines on mental health and psychosocial support in emergency settings includes a lot of important principles and action points related to mental and intellectual ...
  • Incoming 6th grade ppt

    Incoming 6th grade ppt

    6th Grade Schedule. 7 classes-45 minutes in length. 30 minute lunch. 5 minute passing period between classes. 5 core classes (English, Reading Block, Math, Science, Social Studies) 1 elective. P.E. Advisory
  • Hall, Accounting Information Systems

    Hall, Accounting Information Systems

    Malicious & Destructive Programs. Audit objectives: verify effectiveness of procedures to protect against programs such as viruses, worms, back doors, logic bombs, and Trojan horses. Audit procedures: review or verify… training of operations personnel concerning destructive programs. testing of new...
  • Welcome to Valley Gardens SATS information evening 2019

    Welcome to Valley Gardens SATS information evening 2019

    The texts in the English reading booklet will not be linked by a theme. The booklet will contain three or four texts. The least demanding text will come first with the following texts increasing in level of difficulty.