WHEN two expressions are equal to one another, they are written with the sign of equality between them, and the whole is called an equation. Thus x − a=b+c iz ai tion; x — a is called the left side, and be the right male in the equation. REDUCTION. Reduction is the method of bringing the unknown qu to stand alone upon one side of the equation, and the bran quantities upon the other. This is performed by the towe ing rules taken in their order. RULE 1.-If a term be divided by any great-ny, the an every term by the divisor. In this way the equation may be crash dʻvalina RULE 2.-Any term may be trespect, tras ina whe the equation to the other, bygg mags A or from to +. In this way the terms ontzing the with game may be brought to one side of tur kudira terms to the other; after wus tury may one by addition. COR.-If a term be fount in sus rotas de P and it may be erased from voti. RULE 3-If the uni qatı • other, divide both sits in the In this way the tale o when there are unsur RULE 4-the equation it to one side in neet. In the second line the equation is cleared of fractions, and in the third line the quantities 19 and 3x are transposed with their signs changed; and it is evident that the two sides of the equation have been kept equal to one another in every line. 2. Let the equation be (3x+1)+5=10 The removal of the sign from the radical is equivalent to the raising of it to the power. 28. a+x=[a2+x(b2+x2)}]}. • x= 29. 3x-c++ =4x+27 b2 4a abcd x= a (3b+a)d-2ab(2d+1)* EXTERMINATION OF UNKNOWN QUANTITIES. WHEN there are several unknown quantities, there must be as many equations: from these an equation must be deduced, which contains only one of the unknown quantities; and this equation is to be resolved by the preceding rules. This elimination may be performed by any of the following methods: METHOD 1.-Find a value of one of the unknown quantities in each of the equations, supposing all the rest to be known. Make these values equal to one another, and from them find values of another unknown quantity. Make again these values equal, and find another unknown quantity, and so on, until an equation be obtained containing only one unknown quantity, which is to be resolved by the preceding rules. METHOD 2. Find a value of one of the unknown quantities in that equation in which it is least involved; substitute this value and its powers for that unknown quantity and its powers in all the other equations, and proceed in the same way with these equations to get rid of other unknown quantities. METHOD 3.-Multiply the equations by such quantities as will make the coefficients of one of the unknown quantities, or of its highest power, the same in all the equations; then, if the signs of these equal terms be like, subtract the equations, but if the signs be unlike, add them, and new equations will arise, wanting that unknown quantity or its highest power, and these equations are to be treated in the same way. NOTE. The first method seems to be the most regular ; the second is shorter than the first, but the reductions are more intricate; the third is the most simple and expeditious. 1. Let the equations be x+y=12, and 5x+3y=50. 50-3y By Method 1. x=12-y, and x= ; therefore 12-y= 50-3y 5 : whence y=5, and x=12—y=7. By Method 2. 5(12—y)+3y=50 or 60—5y+3y=50: whence y=5, and x = = 7. By Method 3. 5x+5y=60 |