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Solving Linear Systems of Equations with Three Unknowns

Learn three methods for solving a system of three linear equations with three variables: the method of elimination, the method of determinants, and the method of matrices.

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Solving Linear Systems of Equations with Three Unknowns

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  1. Section 5.3 Linear Systems of Equations

  2. THREE EQUATIONS WITH THREE VARIABLES Consider the linear system of three equations below with three unknowns (variables). A solution of this system is simply a triple x, y, z of numbers that satisfy all three equations.

  3. SOLVING A THREE EQUATION SYSTEM We will discuss three methods for solving a three equation system. (These methods are three of the ones discussed in the last section.) • The method of elimination. • The method of determinants. • The method of matrices. NOTE: Graphs are not useful here because the graph of a linear equation with three variables is a plane. Thus, it requires graphing in 3 dimensions.

  4. METHOD OF ELIMINATION Note that there are three pairs of equations: (1) the first and second, (2) the first and third, and (3) the second and third. To use eliminations we • Eliminate x from two of the pairs of equations. (This leaves two equations with variables y andz.) • Use the elimination method for two variables (discussed in Section 5.2) to solve the two resulting equations in Step 1. • Substitute the values of y and z into one of the original equations to find the value for x.

  5. 3 × 3 DETERMINANT To compute the value of the 3 × 3 coefficient determinant:

  6. DETERMINANT METHOD The determinant solution to the system of equations is where Δ is the coefficient determinant.

  7. For the system let Then the solution is MATRIX METHOD

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