Equation procedure

The identification process can be used to solve systems of equations are used. It is relatively easy to use with simple systems of equations.

In the equation procedure, two equations are rearranged so that their left-hand sides are identical and only contain one variable that does not exist on the right-hand sides. Then the two right-hand sides are set equal, so that the newly emerging equation is less dependent on one variable.

example

${\displaystyle {\begin{aligned}&(1)&x+y&=&3\\&(2)&x\cdot y&=&2\end{aligned}}}$

Move

The equations are now rearranged according to a variable, here after . This gives the following equations: ${\displaystyle x}$

${\displaystyle {\begin{aligned}&(1')&x&=&3-y,\\&(2')&x&=&{\frac {2}{y}}.\end{aligned}}}$

Equate

Since the left pages are identical, the same must apply to the right pages. You therefore set these equal and you get an equation that only contains the unknown : ${\displaystyle y}$

${\displaystyle {\begin{aligned}3-y&={\frac {2}{y}}\end{aligned}}}$

Solve the resulting equation

Determine the y-values

${\displaystyle {\begin{aligned}3-y&={\frac {2}{y}}&|&\cdot y\\y(3-y)&=2&|&{\text{ Klammer aufl}}{\ddot {\mathrm {o} }}{\text{sen}}\\3y-y^{2}&=2&|&-3y\\-y^{2}&=2-3y&|&+y^{2}\\0&=y^{2}-3y+2&|&{\text{ quadratische Gleichung in der Normalform}}\rightarrow {\text{L}}{\ddot {\mathrm {o} }}{\text{sen}}\\y_{1}&=1,&&\\y_{2}&=2.&&\end{aligned}}}$

Two solutions for are obtained here , which indicates that the system can also have two pairs of solutions . ${\displaystyle y}$ ${\displaystyle (x\,|\,y)}$

Determine the x-values

The solutions for are inserted into one of the two output equations (or their modified variant) and the corresponding one is calculated from this . ${\displaystyle y}$ ${\displaystyle x}$

${\displaystyle {\begin{aligned}y_{1}:\quad &x_{1}&=&3-y_{1}\\&x_{1}&=&3-1\\&x_{1}&=&2\\y_{2}:\quad &x_{2}&=&3-y_{2}\\&x_{2}&=&3-2\\&x_{2}&=&1\\\end{aligned}}}$

Summary

Thus the system of equations has two solutions : ${\displaystyle (x\,|\,y)}$

${\displaystyle \mathbb {L} =\{(2\,|\,1),(1\,|\,2)\}}$

See also

• Installation procedure
• Addition method

Web links

• The equation procedure in the online math book