SIMPLE EQUATIONS.] 



MATHEMATICS. ALGEBRA. 



471 



6 + 6 . . 



so that x or f- 2 ' = 4 . . x = 4, y= 

 o o 



This is called the method of substitution. 



3. 



3. In the third method, the first object is to convert 

 the proposed equations into forms such that one of the 

 unknown quantities may have the same coefficient in each ; 

 thus : multiplying the first equation by 3, and the 

 second by 2, they become changed into 



6x+15i/=69 



6x 4i/=12 



By subtracting, 1 9t/ = 57 .'. y = 3 



And this value put for y in one of the given equations, 

 in the second for instance, there results, 3j; = 6 . . 



u 



Or without, in this way, borrowing from the method 

 of substitution, multiply the first equation by 2, and the 

 second by 5 : they then change into 



lox 1% = 30 



By adding, 19.t = 70.'.x = 4 



This is the method of equalising tfte coefficient* of the 



8ame unknown quantity in the two equations. When 



this equality is brought about, then addition or sub- 



traction, according as the signs of the equal coefficients 



are unlike or like, will of course remove one of the 



unknown quantities altogether: when one of the un- 



'.vu quantities is thus got rid of, it is said to be 



< filed. 



The operations by the tliree methods may be stated 

 in a rule as follows : 



RULE 1. By equating two erpressions for the same un- 

 y. Find an expression for one of the 

 unknown quantities from the first equation. Find an 

 expression for the same from the second equation. 

 Equate these two expressions, and you will then liave 

 but a single equation with one unknown quantity, the 

 value of which may be found by the former rules. The 

 value of the other is got by substituting the value just 

 found in one of the expressions in place of its symbols. 



2. By substitution. Find an expression for one of 

 the unknown quantities from either equation, and sub- 

 stitute tliis expression in place of that unknown in the 

 other equation: an equation with but one unknown 

 quantity will be the result. 



3. By equalising coefficients. Multiply the two equa- 

 tions by such numbers (or quantities), the smaller the 

 better, as will cause the resulting coefficients of one of the 

 unknown quantities to be the same in the two changed 

 equations : then by addition or subtraction, according as 

 the equal coefficients have unlike or like signs, an equa- 

 tion will arise having only one unknown quantity. Both 

 may be eliminated one after the other in this way ; or 

 Laving eliminated one, and then found the value of the 

 other from the resulting equation, the value of tin; 

 former may then be got by substitution, as in the 

 example above. 



1. Find the rilucj of x and y from the equations 



Multiplying the first equation by 2 and the second by 3, 



n 



Subtracting, ^4 5 = 5 .'. 9.c 



= 60 /. 5x=60 



/. (equation 1) 2 + 



f+y = 7 ].-. clearing, 



= 42-l ...[A] 



Substituting this valuo of x in the first equation, 



.-. 144 



72 ~f 



t.-. 60=5/.-.] 

 3'J 



And cc=24 * =24 18=6. 



2 



Or, subtracting the upper from the lower of [A], 



Substituting this in the former, 

 3x + 12 + 2^=42 .-. 5*=30 .'. x=6 .-. y = G + x = 12. 



This latter is the easier mode of proceeding : such 

 slight departures from rule, as in equations with ona 

 unknown, may often bo adopted with advantage. 



EXAMPLES FOB EXERCISE. 



1. 2.r + 3y = 23 



+ 3y = 23-l 

 -2y = 10J- 



5x 



3 - * + 8y = 

 8~ * 



y + sx 



s ^ 



5. Sx 7y _2* + y J 

 3 5 



2- x -rry=o 



5 



= -"i j*-t 



= i3i] y^=i\ 



6. 



3 



y + 5, 



4 



10* 





7. a.r + Ay = l 1 

 a'jc b'y =- 1 J 



S. ax + tiy = c 1 

 a'x -\- b'y = c J 



I := 12~) Here the unknowns had better be regarded 



as ->- ; when these are found, the reci- 

 x y procals can be taken. 



10. x 2 10 x y 10_ n -, 

 3- -4 



,_?- + lf _'.+ = Of 

 8 4 



11. 



*+n_y+2 , 

 >~y-2 1 



* J LfeZJ I 



J 



12. x-\-y= 19 1 

 x- y a =95J 



y 13 J 



Questions in Simple Equations with One or Two Unknown 

 Quantities. 



1. Find a number such that whether it be divided 

 into two equal parts, or into three equal parts, the 

 product of the parts shall be the samo. 



Let x be the number; then by the question 



2 X 2 



3 X 3 X 3 



27 



. . . " ' , O"T I 



that is -A-;.'. 1 = 07' ' ' 



* ^f v v * 



When tho question implies j>ar& of tho unknown 

 quantity, fractions may generally be avoided in tho 

 solution by representing the unknown not by x, but a 

 multiple of x, such that the proposed parts may bo 

 integral : thus, in the present case 



Let 6x be the number ; then by tho qtiestion , 

 3x X 3x = 2x X 2x X 2* 



9 27., 



that is 



.-. 9=8x .-. x= .'. 



2. A person after spending one-fifth of his income 

 and 10 more, found that he had 35 more than half 

 his income left ; what was his income ? 



Let lOx represent his income ; then by tho question, 

 the number of pounds he spent is 2x, + 10, so that ho 

 had 10x (2x + 10) left ; that is 8x 10 ; but by the 

 question he had 5x + 35 left, 



... % x 10=5x + 35 .-. 3x=45 .'. *=15 .-. 10x = 150, 

 so that his income was 150. 



3. There is a fraction such that if 3 be added to the 

 numerator, its value is J ; and if 1 be subtracted from 

 the denominator, its value is : what is the fraction 1 



. See nte, p. 170. 



