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Solving Systems of Linear Equations in Three Variables

Explore the methods and applications of solving systems of linear equations in three variables using substitution and addition methods. Understand geometric interpretations, inconsistent and dependent systems.

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Solving Systems of Linear Equations in Three Variables

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  1. Chapter 4 Systems of Equations and Inequalities

  2. Chapter Sections 4.1 – Solving Systems of Linear Equations in Two Variables 4.2 – Solving Systems of Linear Equations in Three Variables 4.3 – Systems of Linear Equations: Applications and Problem Solving 4.4 – Solving Systems of Equations Using Matrices 4.5 – Solving Systems of Equations Using Determinants and Cramer’s Rule 4.6 – Solving Systems of Linear Inequalities

  3. Solving Systems of Linear Equations in Three Variables § 4.2

  4. Definitions Theequation 2x – 3y + 4z = 8 is an example of a linear equation in three variables. The solution to this type of equation is an orderedtriple of the form (x, y, z). One possible solutionto the equation 5x – 3y + 4z = 9 is (1, 2, 3).

  5. Solving Systems Systems in three (or more) linear equations are solved the same way systems of two linear equations are solved by using either the substitution or addition method. Solve the following system of equations using the substitution method.

  6. SolvingSystems Since we know that x = -3, we can substitute it into the equation 3x + 4y = 7 and solve for y.

  7. SolvingSystems Now we substitute x=-3 and y=4 into the last equation and solve for z. The solution is the ordered triple (-3, 4, 5).

  8. z 3 (4, 5, 3) y 5 4 x GeometricInterpretation The following is a geometric interpretation of the solution (4, 5, 3).

  9. Inconsistent and Dependent Systems Inconsistent System of Equations A system that has no solution. Example: You obtain a statement that is always false, such as 3=0. Dependent System of Equations A system that has an infinite number of solutions. Example: You obtain a statement that is always true, such as 0=0.

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