Part III: Conquering Conics and Systems of Equations in .NET

Encoding QR Code JIS X 0510 in .NET Part III: Conquering Conics and Systems of Equations

c1 b 2 - b1 c 2 c1 b 2 - b1 c 2 = d a1 b 2 - b1 a 2 a1 c 2 - c 1 a 2 a1 c 2 - c 1 a 2 = d a1 b 2 - b 1 a 2

You can solve most systems of equations with elimination or substitution, but some systems can get pretty gruesome because of the fractions that pop up If you use Cramer s Rule, the work is much easier (see the previous section for the necessary setup) 13x + 7y = 25 with Cramer s Rule, for example, you first find the To solve * 10x - 9y = 13 value of the denominator you ll create, d Using the formula d = a1b2 b1a2 (the subscripts point to which equations the a and b values come from), d = 13( 9) 7(10) = 117 70 = 187 With your d value in tow, try solving for x first: x= c 1 b 2 - b 1 c 2 25 ^ - 9 h - 7 ^13h - 225 - 91 - 316 316 = = = = d - 187 - 187 - 187 187

Gabriel Cramer (1704-1752) was a Swiss mathematician/astronomer/philosopher who modern algebra students recognize for his slick method for solving systems of linear equations Cramer was a bit of a child prodigy, earning his doctorate when he was 18 His thesis was on the theory of sound He aspired to teach philosophy in Geneva but decided to share teaching responsibilities in mathematics with Calandrini instead Cramer taught geometry and mechanics in French rather than the traditional Latin, which was a bit of a departure for the members of the Academy Cramer spent a good deal of time with mathematicians throughout Europe, sometimes editing and other times distilling their work He made contributions to the general knowledge of planets their shapes and positions in their orbits He also published articles on the date of Easter, the history of mathematics, and the aurora borealis and even dabbled in the probability applied to witnesses in a court case His mathematical expertise aided him in his roles in local government, where he advised on fortification, building construction, and artillery If there was ever a case for the practicality of mathematics, Cramer made it

You don t want to substitute such a scary number into one of the equations to solve for y, so move on to the formula for y: y= a 1 c 2 - c 1 a 2 13 ^13h - 25 ^10h 169 - 250 = = = - 81 = 81 d - 187 - 187 - 187 187

Checking your solution isn t much fun because of the fractions, but substituting the values of x and y into both equations shows you that you do, indeed, have the solution If you get d = 0, you have to stop You can t divide by zero The zero value for d indicates that you have either no solution or an infinite number of solutions the lines are parallel, or you have two equations for the same line In any case, Cramer s Rule has failed you, and you have to go back and use a different method

Systems of three linear equations also have solutions: sets of numbers (all the same for each equation) that make each of the equations true When a system has three variables rather than two, you no longer graph the equations as lines To graph these equations, you have to do a three-dimensional graph of the planes represented by the equations containing the three variables In other words, you really can t find the solution by graphing The best method for solving systems with three linear equations involves using your algebra skills