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Systems of Equations Summary Independent • The equations of a linear system are independent if none of the equations can be derived algebraically from the others. Independence • When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. Is this set of equations independent or dependent? x+ y+z = 2 2x - 3y + 2z = 5 3x - 2y + 3z = 7 Is this set of equations independent or dependent? x+ y+z = 2 2x - 3y + 2z = 5 3x - 2y + 3z = 7 Give a geometrical explanation of this system of liner equations. x+ y+z = 2 2x - 3y + 2z = 5 3x - 2y + 3z = 7 The three planes intersect on a line. There are an infinite number of solutions x+ y+z = 2 2x - 3y + 2z = 5 3x - 2y + 3z = 7 Consistent • A linear system is consistent if it has a solution, and inconsistent otherwise. • When the system is inconsistent, it is possible to derive a contradiction from the equations, that may always be rewritten such as the statement 0 = 1. • Write a system of equations that describe 3 parallel planes. • Is the system dependent or independent? • Is the system consistent or inconsistent? Example • Independent and inconsistent x + 2y + 3z = 4 2x + 4y + 6z = 10 2x + 4y + 6z = 12 • Write a system of equations that describe 2 parallel planes cut by a third plane. • Is the system dependent or independent? • Is the system consistent or inconsistent? Example • Independent and inconsistent x + 2y + 3z = 4 2x + 4y + 6z = 10 2x + 3y + 5z = 3 • Write a system of equations that describe 3 planes that form a triangular prism. • Is the system dependent or independent? • Is the system consistent or inconsistent? Example • Independent and inconsistent x + 2y + 3z = 4 2x + 4y + 6z = 10 3x + 6y + 9z = 3 • Write a system of equations that describe 3 planes that intersect on a line. • Is the system dependent or independent? • Is the system consistent or inconsistent? Example • Dependent and consistent with an infinite number of solutions x + 2y + 3z = 4 2x + 4y + 6z = 10 3x + 6y + 9z = 14