Problem by any Learner-Here follow a few Examples that will illuftrate the whole. 615x-10y=100 IX3 715x+24y=372 7-68347-272 8349ly 8 Note, When you multiply any Equation by an abfolute Number, you muft dafh the Figure as in Step the 3d, 5th, 7th, &c. to fhew that they are not Steps multiplied together. 2x+5=160 5x3v=9° Given. Now in order to exterminate x, let the first Equation be multiplied by 2, and the fecond by 5, to make the Coefficients of x alike, there will arife the two following Equations, 10x-6v=180 10x+25v=800 the first of which fubtracted from the fecond, we have 31v=620 which Difference being divided by 31 the Coefficient of v, gives v=20, then by tranfpofing ov in the firit Equation, we have 10x180+6v or 10x=180+. 120..X=30. Let 1x+y=13=a 2x+2=14=b 3y+2=15d 1+3+342x+2y+22=42=a+b+d a+b+d REDUCTION of EQUATIONS. If the Quantities of the Equation be fractional, make them on each fide pure Fractions, then multiply ́them crofs-wife and the Product may be reduced as before. Exam. 2. ———..x-64 by extrading the Square Root 4 X on both fides of the Equation, we have x=8. crofs-wife, we have, nx-nb-ab-ax, and by Tranfpofition -, which Equation being multiplied When any Equation has a Radical Sign on one Side, the other Side muft be raised to the fame Dimenfion. Exam. 1. Letax=b, then by fquaring b, we take off the Radical, and we have axb2, Now in order to find x we muft divide both fides of the Equation by a..x 62 When a Problem hath two Dimenfions, it cannot be anfwered by any of the Methods before laid down; and therefore we muft have recourfe to fome other Method; which is by completing the Square, and is performed by the following Rule Add the Square of half the Co-efficient of the unknowu quantity to both fides of the Equation, and the Square will be complete, Given x2+2ax=b Required x. added as above; we have x2+zax+a2—a2+b, Then by extracting the root on both Sides of the Equation we have x+ava2+b and by tranf pofing a we have xa2+D-a. of L Of PROBLEMS. 1. Problems are Queftions to be folved. 2. The Solution of a Problem is, the Anfwer to a Queftion, or the Determination of the quantity fought. 3. The Problem has oftentimes various Answers, and therefore it is neceffary to know when it is truly limited; which may be known by the following Rule. When the number of Quantities fought, exceeds the Number of Equations given, the Question admits of va rious answers. Suppofe x+y=20 now it is plain x may be any Number, whole or broken, being less than 20, and y the Remainder. Rule. When the Number of Quantities fought are equal the number of Equations (not depending on each other) the Queftion is truly limited. Axioms for the more ready folving of Questions. Ax. 1. If from the Sum of any two Quantities either Quantity be taken, the remainder is the other Quantity. Ax. 2. The difference of any two Quantities being added to the lefs, the Sum is the greater. Ax. 3. The Product of any two Quantities being divided by either Quantity, the Quotient is the other. Ax. 4. The Quotient of any two Quantities being multiplied by the lefs, the Product is the greater. Ax. 5. The Rectangle of the Sum, and Difference of any two Quantities, is equal to the difference of their Squares. Ax. 6. The Difference of the Squares of the Sum, and difference of any two Quantities, is equal four times their Rectangle. Ax. 7. The Sum of the Squares of the Sum, and difference of any two Quantities, is equal twice the Sum of -their Squares. y x+y2 Sum of the Squares. x2 y2-Difference of the Squares. Note, The above being understood, any Problem may be easily taken in the Algebraic Method. SOLUTION OF PROBLEMS. Prob 1. The fum of Two Numbers being 14, and their difference 4, required the Numbers, with a Theorem for all fuch Questions? Let 14 a 4b and x=lefs, than b+x will be the greater number per Ax. 2. 12x+ba P.Q. a-b b+x=9 N. B. I fhall follow the above Numbers for 10 Questions. Prob. 2. The Sum of two Numbers za and their Product =p. 2√ Required the Number? x-Difference of the Numbers a+x=greater a- -x-lefs Ia -x=P 2a2-p=xz 3x=√a2-p2=2. ̊.a+x=9, a—x=5 Prob. 3. The Sum of two Numbers 14 (a) and their Quotient 1, 8 (q) required the Numbers. |xlefs & qx=greater Number 1x+qxa P.Q. Prob. 4. The Sum of any two Numbers being 14, and the Sum of their Squares 106, required the Numbers? 1142a, x=1 Difference, 106-b 12a2+2x2-b Prob. 5. The Sum of two Numbers 14, and the Dif ference of their Squares 56, required the Numbers? I x Differ. of the Numbers, 14=2a,56 b 4ax-b Prob. 6. The Difference of two Numbers 4, and their Product 45, required the Numbers ? 2 xSum, 4=2a, 45—b. x+a=greater, a—x—iess a2x2-b x=/a2-b2 x+a=9 | x-a=5 Prob. 7. The Difference of two Numbers being 4, and their quotient 1, 8 required the Numbers? x=lefs, 1, 89, 4b qx=greater qx-x=b b =5 19x=9 Frob. 8. The Difference of two Numbers 4, and the Sum of their Squares 106, required the Numbers? X |