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Except for Bézout's theorem, the general approach was to eliminate variables for reducing the problem to a single equation in one variable. The case of linear equations was completely solved by Gaussian elimination , where the older method of Cramer's rule does not proceed by elimination, and works only when the number of equations equals the ...
An example from mathematics says that a single-variable quadratic polynomial has a real root if and only if its discriminant is non-negative: [1] ∃ x ∈ R . ( a ≠ 0 ∧ a x 2 + b x + c = 0 ) a ≠ 0 ∧ b 2 − 4 a c ≥ 0 {\displaystyle \exists x\in \mathbb {R} .(a\neq 0\wedge ax^{2}+bx+c=0)\ \ \Longleftrightarrow \ \ a\neq 0\wedge b^{2 ...
If the subexpressions are not identical, then it may still be possible to cancel them out partly. For example, in the simple equation 3 + 2y = 8y, both sides actually contain 2y (because 8y is the same as 2y + 6y). Therefore, the 2y on both sides can be cancelled out, leaving 3 = 6y, or y = 0.5. This is equivalent to subtracting 2y from both sides.
Since all the inequalities are in the same form (all less-than or all greater-than), we can examine the coefficient signs for each variable. Eliminating x would yield 2*2 = 4 inequalities on the remaining variables, and so would eliminating y. Eliminating z would yield only 3*1 = 3 inequalities so we use that instead.
One sees the solution is z = −1, y = 3, and x = 2. So there is a unique solution to the original system of equations. So there is a unique solution to the original system of equations. Instead of stopping once the matrix is in echelon form, one could continue until the matrix is in reduced row echelon form, as it is done in the table.
One of the basic principles of algebra is that one can multiply both sides of an equation by the same expression without changing the equation's solutions. However, strictly speaking, this is not true, in that multiplication by certain expressions may introduce new solutions that were not present before. For example, consider the following ...
This implies that a monomial containing an X-variable is greater than every monomial independent of X. If G is a Gröbner basis of an ideal I for this monomial ordering, then G ∩ K [ Y ] {\displaystyle G\cap K[Y]} is a Gröbner basis of I ∩ K [ Y ] {\displaystyle I\cap K[Y]} (this ideal is often called the elimination ideal ).
For example, taking the statement x + 1 = 0, if x is substituted with 1, this implies 1 + 1 = 2 = 0, which is false, which implies that if x + 1 = 0 then x cannot be 1. If x and y are integers, rationals, or real numbers, then xy = 0 implies x = 0 or y = 0. Consider abc = 0. Then, substituting a for x and bc for y, we learn a = 0 or bc = 0.