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In algebra, we often have to solve simultaneous differential equations. If the equations are linear and independent, there is a unique solution when the number of equations equals the number of variables. If there are only two variables and two equations, it's easy, but as the number of equations and unknowns increases, the problem becomes more difficult. Imagine, for example trying to solve 100 equations in 100 unknowns. That's an unthinkably difficult manual exercise, but a good computer program can do it in less than a second. Suppose you want the solution to this pair of equations: 2 * x + y = 1 4 * y = 12 In the first equation, the coefficient of x is 2 and the coefficient of y is 1. In the second equation, the coefficient of x is 0, and the coefficient of y is 4. The right-side values are 1 and 12, respectively. It's easy to solve this pair of equations by hand. The second equation says y = 12/4 = 3, and substituting this back into the first equation gives x = (1-3)/2 = -1.
The length and width of each rectangle are whole numbers. How many different pairs of lengths and widths are there? List them.
Use compound interest to find the amount.
Suppose K is a Galois extension of F of degree p^n for some prime p and some n >= 1. Show that there are Galois extensions of F contained in K of degree p and p^(n-1).
counting principles please respond to the followingcompare and contrast permutations and distinguishable permutations.
Determine the perfect trinomial
Use the previous fact to show that Fermat's equation x^n + y^n = z^n has no integer solutions when n>/= 2 is even and both x and y are odd,
Simplify each expression using the rules of exponents and examine the steps you are taking
Finding permutation
What is the length of BF, in centimeters and how many exams has Jeff taken so far and what is the total area of the surface of the cube that is NOT red?
What is the discriminant? When do you use quadratic formula to solve a quadratic equation? What square root property is essential to solve any radical equation involving square root?
Find the domain of the function and Solve the inequality
costs in an operation costing system are accumulated bya department.b by individual job.c by both job and departments.d
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