All systems from three equations with three unknown are solved in one way – by consecutive replacement of the unknown with the expression comprising other two unknown, reducing thus their number.
Instruction
1. To understand as the algorithm of replacement of unknown works, quality of an example we will take the following system of the equations with three unknown x, y and z: 2x+2y-4z=-124x-2y+6z=366x-4y-2z=-16
2. In the first equation transfer everything composed except x, increased by 2, in the right part and divide into the multiplier standing before x. Thus you receive value x, expressed through two other unknown z and y. x =-6-y+2z.
3. Now work with the second and third equations. Replace everything x with the received expression containing only unknown z and y.4 * (-6-y+2z) - 2y+6z=366 * (-6-y+2z) - 4y-2z=-16
4. Remove the brackets, considering signs before multipliers, perform operations of addition and subtraction in the equations. Transfer composed without unknown (number) in the right member of equation. You receive a system from two linear equations with two unknown. - 6y+14z=60-10y+10z=20.
5. Now allocate unknown y that it could be expressed through z. It is not obligatory to do it in the first equation. On an example it is visible that multipliers at y and z coincided except for the sign therefore work with this equation, it will be so more convenient. Transfer z with a multiplier to the right member of equation and divide both parts into y multiplier - 10.y=-2+z.
6. Substitute the received expression of y in the equation which was not involved, remove the brackets, considering the sign of a multiplier, make actions of addition and subtraction, and you receive:-6 * (-2+z) +14z=6012-6z+14z=608z=48z=6.
7. Now return to the equation where y is defined by z, and put value z in the equation. At you it will turn out: y=-2+z=-2+6=4
8. Remember the very first equation in which x it is expressed through z y. Substitute in it their numerical values. At you it will turn out: x=-6-y+2z=-6-4+12=2Таким in the way, all unknown are found. In just the same way the nonlinear equations where mathematical functions act as multipliers are solved.