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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]
One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
Vertical line of equation x = a Horizontal line of equation y = b. Each solution (x, y) of a linear equation + + = may be viewed as the Cartesian coordinates of a point in the Euclidean plane. With this interpretation, all solutions of the equation form a line, provided that a and b are not both zero. Conversely, every line is the set of all ...
Similarly, if c 1 = 1 / 2i and c 2 = − 1 / 2i , then the independent solution formed is y 2 (x) = e ax sin bx. Thus by the superposition principle for linear homogeneous differential equations , a second-order differential equation having complex roots r = a ± bi will result in the following general solution:
If two solutions intersect each other, that is, they both go through the same point (x,y), then there is a failure of uniqueness for a first-order ordinary differential equation. Thus, there will be a failure of uniqueness if a solution of the first form intersects the second solution. The condition of intersection is : y s (x) = y c (x). We solve
The graph of the zero polynomial, f(x) ... x 3 y 2 + 7x 2 y 3 − 3x 5 is homogeneous of degree 5. ... and, if it is, solve it.
A linear multistep method is zero-stable if all roots of the characteristic equation that arises on applying the method to ′ = have magnitude less than or equal to unity, and that all roots with unit magnitude are simple. [2] This is called the root condition [3] and means that the parasitic solutions of the recurrence relation will not grow ...
or, after expansion and collecting terms with equal powers of x: ( A + B + C ) x 2 − ( 3 A + 2 B + C ) x + 2 A = 1. {\displaystyle (A+B+C)x^{2}-(3A+2B+C)x+2A=1.\,} At this point it is essential to realize that the polynomial 1 is in fact equal to the polynomial 0 x 2 + 0 x + 1, having zero coefficients for the positive powers of x .