At the solution of systems of two equations with two variables usually there is a need to simplify an initial system and by that to lead it to a look, more convenient for the decision. For this purpose reception of expression of one variable through another is often used.
Instruction
1. Transform one of the system equations to a look in which y is expressed through x or, on the contrary, x through y. Substitute the received expression instead of y (or instead of x) in the second equation. You receive the equation from one variable.
2. For the solution of some systems of the equations it is required to express both variables x and y through one or two new variables. For this purpose enter one variable m only for one equation or two variables m and n for both equations.
3. Example of I. Express one variable through another in the system of the equations: │x–2y=1, │x²+xy–y²=11. Transform the first equation of this system: transfer a monomial (– 2y) in the right part of equality, having changed the sign. From here receive: x=1+2y.
4. Substitute in the equation x²+xy–y²=11 instead of x expression 1+2y. The system of the equations will take a form: │(1+2y)²+(1+2y) y–y²=11, │x=1+2y. The received system is equivalent to initial. You expressed variable x in this system of the equations through y.
5. Example of II. Express one variable through another in the system of the equations: │x²-y²=5, │xy=6. Transform the second equation of a system: divide both members of equation of xy=6 into x≠0. From here: y=6/x.
6. Substitute the received expression in the equation x²–y²=5. You receive a system: │x²– (6/x)²=5, │y=6/x. The last system is equivalent to initial. You expressed variable y in this system of the equations through x.
7. Example of III. Express variables y and z through new variables m and n: │2/(y+z) +9 / (2y+z) =2; │4/(y+z) =12 / (2y+z)-1. Let 1 / (y+z) =m and 1 / (2y+z) =n. Then the system of the equations will look as follows: │2/m+9/n=2, │4/m=12/n–1. You expressed variables y and z in the initial system of the equations through new variables m and n.