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2) = 4. Now (3) 3 = 4 satisfies (1), but FRACTIONS ART. 24] x = 3 does not satisfy meaning when x = 3. (1) since 23 the hand member has no left Hence the extraneous root x = 3 is introduced in clearing of fractions. The above example shows the importance of checking each solution by substituting the original equation. Exercises Solve the following equations and check the results: 1" a; 5* o "a? a; g + 4 = 2 g 3 a; _ 2 _______ x l 9x _. - x 5 z = 2 ' 5 .. 8 - 10 3* -- x __ 2. 7. 4 3 u" 3 -| *+ 17 +3 x - m+ m 7 x a: z-2 1 A Q , i - s - x 8 m+ m n n - 9 CHAPTER V FUNCTIONS 25.

2) + y. (3) Method. we have = x Substituting this value for x in 4 +y- Substituting 6 for y in x (1), - = 6 we (2), =- 4y -3y = - or 4 14, y 18, find we find 4, or (4) = 6. x = 10. Hence the required values for x and y are 10 and 6 respectively. This method is known as elimination by substitution. Solution. From Second Method. (1) subtract (2) and we get 3y Multiplying (1) by 4 and (4) and (5), Hence the required (2) = y 18, by 6. (3) 1, the two equations become - 4y = 16, (4) -s + 4y = 14. (5) 4x Adding = we get 3x = solution 30, is x x = = 10.

Let X'X and Y'Y (Fig. 11) be two straight lines meeting at right angles. Let them be tional relations may considered as two number scales having the point of intersection be any point in the plane. as the zero point of each. Let to two lines. Let x represent the From drop perpendiculars A A AN INTRODUCTION TO MATHEMATICS 30 [CHAP. If the distance to F'F, and y the distance to X'X. the left of ', x is considered negative and if A A lies V to below X'X, y is considered negative. It is evident that no matter where A lies in the plane there corresponds to it two and only two numbers and those numbers are the perpendiculars to Y'Y and X'X respectively.

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