874 CHAPTER 9 Systems and Matrices Using graphs to find the solution set of a linear system in two unknowns provides a good visual perspective, but it may be inefficient when the solution set contains non-integer values. Thus, we introduce two algebraic methods for solving systems with two unknowns: substitution and elimination. Linear Systems The definition of a linear equation can be extended to more than one variable. Any equation of the form a1 x1 +a2 x2 +P+an xn =b, for real numbers a1, a2, c , an (all nonzero) and b, is a linear equation, or a first-degree equation, in n unknowns. A set of equations considered simultaneously is a system of equations. The solutions of a system of equations must satisfy every equation in the system. If all the equations in a system are linear, the system is a system of linear equations, or a linear system. 9.1 Systems of Linear Equations ■ Linear Systems ■ Substitution Method ■ Elimination Method ■ Special Systems ■ Application of Systems of Equations ■ Linear Systems with Three Unknowns (Variables) ■ Application of Systems to Model Data Three Cases for Solutions of Linear Systems in Two Variables Case 1 The two graphs intersect in a single point. The coordinates of this point give the only solution of the system. Because the system has a solution, it is consistent. The equations are not equivalent, so they are independent. See Figure 1(a). Case 2 The graphs are parallel lines. There is no solution common to both equations, so the solution set is ∅ and the system is inconsistent. Because the equations are not equivalent, they are independent. See Figure 1(b). Case 3 The graphs are the same line—that is, they coincide. Because any solution of one equation of the system is a solution of the other, the solution set is an infinite set of ordered pairs representing the points on the line. This type of system is consistent because there is a solution. The equations are equivalent, so they are dependent. See Figure 1(c). x y 0 Infinite number of solutions x y 0 No solution x y 0 One solution Consistent system Independent equations Inconsistent system Independent equations Consistent system Dependent equations (a) (b) (c) Figure 1 Substitution Method In a system of two equations with two variables, the substitution method involves using one equation to find an expression for one variable in terms of the other, and then substituting this expression into the other equation of the system.
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