The system of the linear equations contains the equations in which all unknown contain in the first degree. There are several ways of the solution of such system.
Instruction
1. Method of substitution or consecutive exception. Substitution is used in a system with a small amount of unknown. It is the simplest method of the decision for simple systems. At first from the first equation we express one unknown through others, we substitute this expression in the second equation. We express the second unknown from the changed second equation, we substitute received in the third equation, etc. until we calculate the last unknown. Then we substitute its value in the previous equation and we learn penultimate unknown, etc. Let's consider example systems with two unknown.x + y - 3 = 02x - y - 3 = 0vyrazim from the first equation x: x = 3 - y. Let's substitute in the second equation: 2(3 - y) - y - 3 = 06 - 2y - y - 3 = 03 - 3y = 0y = 1podstavlyaem in the first equation of a system (or in expression for x that the same): x + 1 - 3 = 0. Let's receive that x = 2.
2. Method of term by term subtraction (or additions). This method often allows to reduce time of the solution of a system and to simplify calculations. It is in that having analyzed coefficients at unknown thus to put (or to subtract) the system equations to exclude a part of unknown from the equation. Let's review an example, we will take the same system, as in the first method.x + y - 3 = 02x - y - 3 = 0legko to see that at y there are coefficients, identical on the module, but with the different sign, therefore if we put two equations term by term, then it will be possible to exclude y. Let's execute addition: x + 2x + y - y - 3 - 3 = 0 or 3x - 6 = 0. Thus, x = 2. Having substituted this value in any equation, we will find y. It is possible to exclude, on the contrary, x. Coefficients at x are identical according to the sign therefore we will subtract one equation from another. But in the first equation coefficient at x - 1, and in the second - 2 therefore just subtraction will not possible to exclude x. Let's increase the first equation by 2, we will receive such system: 2x + 2y - 6 = 02x - y - 3 = 0teper we will term by term subtract the second from the first equation: 2x - 2x + 2y - (-y) - 6 - (-3) = 0 or, having given similar, 3y - 3 = 0. Thus y = 1. Having substituted in any equation, we will find x.