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The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
The solution set for the equations x − y = −1 and 3x + y = 9 is the single point (2, 3). A solution of a linear system is an assignment of values to the variables ,, …, such that each of the equations is satisfied. The set of all possible solutions is called the solution set. [5]
[a] This means that the function that maps y to f(x) + J(x) ⋅ (y – x) is the best linear approximation of f(y) for all points y close to x. The linear map h → J(x) ⋅ h is known as the derivative or the differential of f at x. When m = n, the Jacobian matrix is square, so its determinant is a well-defined function of x, known as the ...
In binary (base-2) math, multiplication by a power of 2 is merely a register shift operation. Thus, multiplying by 2 is calculated in base-2 by an arithmetic shift. The factor (2 −1) is a right arithmetic shift, a (0) results in no operation (since 2 0 = 1 is the multiplicative identity element), and a (2 1) results in a left arithmetic shift ...
If y 2 = x 3 − x − 1, then the field C(x, y) is an elliptic function field. The element x is not uniquely determined; the field can also be regarded, for instance, as an extension of C(y). The algebraic curve corresponding to the function field is simply the set of points (x, y) in C 2 satisfying y 2 = x 3 − x − 1.
If y = x 3 and y ≠ 0 then each point (x,y) has exactly two tangent lines to γ passing through it (this corresponds to the cubic having one ordinary root and one repeated root). The same is true if y ≠ x 3 and y = 0. If y = x 3 and x = 0, i.e., x = y = 0, then this point has a single tangent line to γ passing through it (this corresponds ...
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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...