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

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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 0 and 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 1 and 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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