Unit 5 Section 2 - 5.2 Systems of Linear Equations in Three...

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5.2 Systems of Linear Equations in Three Unknowns Objectives: After completing this section, you should be able to: 1. Solve linear systems in three unknowns using Gaussian elimination 2. Use systems of linear equations in three unknowns to model and solve application problems Keep This in Mind A linear equation in three variables is an equation that can be put in the form d cz by ax where x , y , and z are the variables, d c b a and , , , are real numbers; and c b a and , , are not all zero at the same time. The number a is the leading coefficient , and x is the leading variable. The graph of a linear equation in three variables is a plane in a three dimensional coordinate system. An example of what a plane would look like is a floor or a desk top. In general, we are interested in solving linear systems in three variables of this type 3 3 3 3 2 2 2 2 1 1 1 1 d z c y b x a d z c y b x a d z c y b x a where 3 2 1 3 2 1 3 2 1 3 2 1 and , , , , , , , , , , , d d d c c c b b b a a a are all real numbers. By a solution to a system of three linear equations in three variables we mean an ordered triple that makes ALL three equations true. Geometrically, it is where the graphs of the three equations intersect, what they have in common. Just as with linear systems in two variables, there are three possible outcomes that you may encounter when working with systems of linear equations in three variables: 1. exactly one or unique solution 2. no solution 3. infinitely many solutions The terms consistent , inconsistent , dependent , and independent are used to describe these solutions, just as they are for linear systems with two variables.
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Unit 5. Systems of Equations and Inequalities Section 2 page 2 _____________________________________________________________________________________ Solving Systems of Linear Equations in Three Variables There is more than one way that you can solve this type of systems. But we will only look at solving them using the more preferred way, elimination by addition, specifically using Gaussian elimination, a more systematized method for solving linear systems. This method is just an extension of what you learned in linear systems in two variables. The following are two examples of systems of linear equations in three variables. The second system is in upper triangular form said to be in row-echelon form , which means a “stair-step” pattern with leading coefficients of 1. Note that the leading variable in the 1 st equation is x , in the 2 nd equation is y , and in the 3 rd equation is z. A system of linear equations A system in triangular form ( row-echelon form ) Gaussian elimination is the process of solving a system of linear equations by transforming the original system to an equivalent system that is in row-echelon form by using elementary equation operations and back-substitution from Section 5.1.
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This note was uploaded on 05/13/2011 for the course MATH 17 taught by Professor Dikopaalam during the Spring '11 term at University of the Philippines Los Baños.

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Unit 5 Section 2 - 5.2 Systems of Linear Equations in Three...

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