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Section 3.1 Solving Systems of Linear Equations by the Graphing Method 237 y 1 Figure 3-1 x 3x y 2 54 3 1 2 3 4 5 2 3 4 5 6 7 2 1 (2,4) 1 3 2 x + y = 6 Graphing the lines from Example 1 we see that the point of intersection is 12,42. Therefore, we say that the solution set is 512,426. See Figure 3-1. When two lines are drawn in a rectangular coordinate system, three geometric relationships are possible: 1. Two lines may intersect at exactly one point. 2. Two lines may intersect at no point.This occurs if the lines are parallel. 3. Two lines may intersect at infinitely many points along the line.This occurs if the equations represent the same line (the lines are coinciding). If a system of linear equations has one or more solutions, the system is said to be a consistent system. If a linear system has no solution, it is said to be an inconsistent system. If two equations represent the same line, then all points along the line are solutions to the system of equations. In such a case, the equations are said to be dependent. If two linear equations represent different lines, then the equations are said to be independent. The different possibilities for solutions to systems of linear equations are given in Table 3-1. Table 3-1 Solutions to Systems of Linear Equations in Two Variables One Unique Solution No Solution Infinitely Many Solutions One point of intersection Parallel lines Coinciding lines System is consistent. System is inconsistent. System is consistent. Equations are independent. Equations are independent. Equations are dependent. 2. Solving Systems of Linear Equations by Graphing Solving a System of Linear Equations by Graphing Solve the system by graphing both linear equations and finding the point(s) of intersection. y 1 2 x 2 4x 2y 6 Example 2


miller_intermediate_algebra_4e_ch1_3
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