Search results
Results From The WOW.Com Content Network
The following examples of indeterminate systems of equations have respectively, fewer equations than, as many equations as, and more equations than unknowns: System 1: x + y = 2 {\displaystyle {\text{System 1: }}x+y=2}
Consider the system of 3 equations and 2 unknowns (X and Y), which is overdetermined because 3 > 2, and which corresponds to Diagram #1: = = = + There is one solution for each pair of linear equations: for the first and second equations (0.2, −1.4), for the first and third (−2/3, 1/3), and for the second and third (1.5, 2.5).
If a system of equations is inconsistent, then the equations cannot be true together leading to contradictory information, such as the false statements 2 = 1, or + = and + = (which implies 5 = 6). Both types of equation system, inconsistent and consistent, can be any of overdetermined (having more equations than unknowns), underdetermined ...
Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown. Repeat steps 1 and 2 until the system is reduced to a single linear equation. Solve this equation, and then back-substitute until the entire solution is found. For example, consider the following system:
For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1.
In mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns [1] (in contrast to an overdetermined system, where there are more equations than unknowns).
In the case of n equations in n unknowns, it requires computation of n + 1 determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single determinant. [8] [9] [verification needed] Cramer's rule can also be numerically unstable even for 2×2 systems. [10]
A trigonometric equation is an equation g = 0 where g is a trigonometric polynomial. Such an equation may be converted into a polynomial system by expanding the sines and cosines in it (using sum and difference formulas), replacing sin(x) and cos(x) by two new variables s and c and adding the new equation s 2 + c 2 – 1 = 0.