SECTION 11.1 Systems of Linear Equations: Substitution and Elimination 761 Figure 5 illustrates the system given in Example 8 using GeoGebra. Notice that the graphs of the two equations are lines, each with slope −2 and each with y-intercept 4. The lines are coincident, and the solutions of the system, such as ( ) ( ) −1, 6 , 0, 4 , and ( ) 2, 0 , are points on the line. Notice also that equation (2) in the original system is −3 times equation (1), indicating that the two equations are dependent. Now Work PROBLEMS 27 AND 31 5 Solve Systems of Three Equations Containing Three Variables Just like a system of two linear equations containing two variables, a system of three linear equations containing three variables has • Exactly one solution (a consistent system with independent equations) • No solution (an inconsistent system) • Infinitely many solutions (a consistent system with dependent equations) Solving a system of three linear equations containing three variables can be viewed as a geometry problem.The graph of each equation in such a system is a plane in space. A system of three linear equations containing three variables represents three planes in space. Figure 6 illustrates some of the possibilities. Solution (1) (2) (1) Multiply equation (1) by 3. (2) Add equations (1) and (2). Figure 5 y x2 4 =− + Figure 6 Consistent system; one solution Solution Solutions Solutions (a) Consistent system; infinite number of solutions (b) Inconsistent system; no solution (c) We choose to use the method of elimination. x y x y 2 4 6 3 12 + = − − =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 6 3 12 6 3 12 + = − − =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ 0 0 = The statement = 0 0 means the original system is equivalent to a system containing one equation, + = x y 2 4. So the equations of the system are dependent. Furthermore, any values of x and y that satisfy + = x y 2 4 are solutions. For example, = x 2, = = = = − = = = − y x y x y x y 0; 0, 4; 1, 6; 4, 4 are solutions. There are, in fact, infinitely many values of x and y for which + = x y 2 4, so the original system has infinitely many solutions. We write the solution of the original system either as = − + y x x 2 4, where can be any real number or as = − + x y y 1 2 2, where can be any real number The solution can also be expressed using set notation: x y y x x x y x y y , 2 4, any real number or , 1 2 2, any real number { } {( ) } ( ) = − + = − +
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