unknown quantity, we will explain in what manner, before we proceed to a general solution of equations containing two unknown quantities. Let the price of a pair of shoes. Then 3x= the price of three pair of shoes. And 16-3x- the price of two pair of boots. Consequently 16-3x = the price of one pair of boots. Now 4 pair of shoes which cost 4x, and 3 pair of boots which cost 48-9x 2 being added together, must equal 23 dollars. That is, 4x-24-3x=23. Or, of a pair of shoes. 16-3x 2 1-x=0. Therefore x-2 dollars, the price Substitute the value of x in the expression and we find 5 dollars for the price of a pair of boots. Now let us resume the equations, 3x+2y=16 (1) FIRST METHOD OF ELIMINATION. (Art. 46.) Transpose the terms containing y to the right hand sides of the equations, and divide by the coefficients of x, and From equation (A) we have x= 16-2y (C) And from (B) we have X= 23-3y (D) Put the two expressions for x equal to each other. (Ax. 7.) An equation which readily gives y=5, which, taken as the value of y, in either equation (C) or (D) will give x=2. This method of elimination, just explained, is called the method by comparison. SECOND METHOD OF ELIMINATION. (Art. 47.) To explain another method of solution, let us again resume the equations : The value of x from equation (A) is x=(16—2y). Substitute this value for x in equation (B), and we have 4X (16-2y)+3y=23, an equation containing only y. Reducing it, we find y=5 the same as before. This method of elimination is called the method by substitution, and consists in finding the value of one unknown quantity from one equation to put that value in the other which will cause one unknown quantity to disappear. THIRD METHOD OF ELIMINATION. (Art. 48.) Resume again 3x+2y=16 4x+3y=23 (A) When the coefficients of either x or y are the same in both equations, and the signs alike, that term will disappear by subtraction. When the signs are unlike, and the coefficients equal, the term will disappear by addition. To make the coefficients of x equal, multiply each equation by the coefficient of x in the other. To make the coefficients of y equal, multiply each equation by the coefficient of y in the other. Multiply equation (A) by 4 and 12x+8y=64 Multiply equation (B) by 3 and 12x+9y=69 Difference y=5 as before. To continue this investigation, let us take the equations Multiply equation (A) by 2, and equation (B) by 3, and we Equations in which the coefficients of y are equal, and the signs unlike. In this case add, and the y's will destroy each other, giving Or 19x=76 x=4. This method of elimination is called the method by addition and subtraction. FOURTH METHOD OF ELIMINATION. (Art. 48.) Take the equations 2x+3y=23. (A) Multiply one of the equations, for example (A), by some indeterminate quantity, say m. Then Subtract (B) Remainder, (C) (2m−5)x+(3m+2)y=23m-10 As m is an indeterminate quantity, we can assume it of any value to suit our pleasure, and whatever the assumption may be, the equation is still true. Let us assume it of such a value as shall make the coefficient of y, (3m+2)=0. The whole term will then be 0 times y, which is 0, and equation (C) becomes (2m-5)x=23m-10 Or x= 23m-10 (D) But 3m+2=0. Therefore m=— -3. Which substitute for m in equation (D), and we have -23X-10 -23×2-30 -76 =4. 19 This is a French method, introduced by Bezout, but it is too indirect and metaphysical to be much practised, or in fact much known. Of the other three methods, sometimes one is preferable and sometimes another, according to the relation of the coefficients and the positions in which they stand. No one should be prejudiced against either method, and in practice we use either one, or modifications of them, as the case may require. The forms may be disregarded when the principles are kept in view. (Art. 49.) To present these different forms in the most general manner, let us take the following general equations, as all particular equations can be reduced to these forms. Observe that a and a', may represent very different quantities, so b and b' may be different, also c and c' may be different. In special problems, however, a may be equal to a', or be some multiple of it; and the same remark may apply to the other letters. In such cases the solution of the equations are much easier than by the definite forms. Hence, in solving definite problems great attention should be paid to the relative values of the coefficients. First method. Transpose the terms containing y and divide by the coefficients of x, and Clearing of fractions, give a'c-a'by-ac'-ab'y. c_by_c (Axiom 7.) Transpose, and (ab'-a'b)y=ac-a'c. ac'-a'c By division y=ab'—a'b' When y is determined, its value put in either equation marked |