Chapter7 - : Pre-Calculus - Chapter 7A Chapter 7A - Systems...

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© : Pre - Calculus - Chapter 7A Chapter 7A - Systems of Linear Equations Geometry of Solutions In an earlier chapter we learned how to solve a single equation in one unknown. The general form of such an equation has the form ax b , where the constants a 0and b are assumed known, and we are looking for a value of x which satisfies the equation. Since a 0, the equation is easy to solve. Multiply by a 1 . Thus, we have the solution x b a . The situation is not quite so simple when we have more than one equation and unknown. First we give the standard form for a system of two equations in two unknowns. ax by c dx ey f , where the constants a , b , c , d , e ,and f are assumed known. By a solution of this system we mean a pair of numbers x 0 and y 0 which satisfy the system of equations. That is, when the substitutions x x 0 and y y 0 are made in the system both equations become identities. This pair of numbers is commonly written as x 0 , y 0 and interpreted as a point in the Euclidean plane, R 2 . By the solution set of a system we mean the totality of all possible solutions to the system. Example 1 : Which of the pairs of numbers 1,2 , 11,3 ,or 19, 9 are solutions to the given system? 2 x 5 y 7 x 2 y 1. Solution: To check that a pair of numbers is a solution we substitute the values in for x and y .In the first equation if we substitute x 1and y 2 we have 2 x 5 y 7 2 1 5 2 7 8 7 Since 8 7 the first pair does not satisfy the first equation let alone both equations. We try the second pair 11,3 next: 2 x 5 y 7 2 11 5 3 7 7 7 Okay, the second pair satisfies the first equation, but that is not enough to guarantee that we have a solution to the system. The second equation still has to be checked, which we do below. x 2 y 1 11 2 3 1 5 1 Since 5 1, this pair does not solve the system. © : Pre - Calculus
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© : Pre - Calculus - Chapter 7A Now let’s check the last pair 19, 9 to see if it is a solution 2 x 5 y 7 2 19 5 9 7 7 7 2 x 5 y 7 2 19 5 9 7 7 7 Since the pair 19, 9 satisfies both equations, this pair is a solution to the system. Question : Does the pair of numbers 1, 1 satisfy the system? 2 x y 3 3 x 6 y 3 Answer: No, the first equation is satisfied, but the second is not. Note: There is no reason why a system must consist of two equations. In the following pages we will have examples of systems which consist of a single equation with more than one unknown, systems which consist of two equations with three unknowns, and finally the general system which consists of n equations in m unknowns. In this later case there need be no a-priori relationship between
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This note was uploaded on 02/23/2010 for the course CHEM 107 taught by Professor Generalchemforeng during the Fall '07 term at Texas A&M.

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Chapter7 - : Pre-Calculus - Chapter 7A Chapter 7A - Systems...

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