Systems of equations solving systems of linear equations

The concept of the system and its rozvytku

Definition: If the task is to find all common solutions of two (or more) equations with one or more variables, then we say that it is necessary razvesti system of equations.

Definition: Rozvyazka system — the value of a variable or such an ordered set of values smya that satisfies all the equations of the system, i.e. rozvyazka system of two or more equations with unknown call such an ordered set a lot of numbers, after the substitution of which into the system is unknown, all the equations become true numerical equality.

Definition: Razvesti the system of equations is to find all its interchanges , or to prove that they are not.

If the system has no solution, then it is incompatible.

Examples of systems

— a system of two equations with two variables

Pair that is a solution of the system

— the system of three equations with three variables

Three that is one of rozvytku system

Scheme the solution of systems of equations

The graphical method

  1. Perform equivalent transformations, so that it is convenient to graph the function. For example:
  2. Build graphics.
  3. Find the point of intersection of the graphs. The coordinates of these points is rozvyazka this system of equations.

The substitution method

  1. One equation of the system we Express one variable via another, always choose a convenient variable. For example, from the equations we Express the variable and not Vice versa.
  2. Substitute the value found into another equation of the system and obtain an equation with one variable.
  3. Rozvadov the derived equation
  4. The value found is substituted in the expressed equation and find the value of the second variable.

Method of adding

  1. Urunov the coefficients of one variable by term multiplication of both equations on multipliers, selected accordingly.
  2. Add (or subtract) pocino two equations of the system, thereby eliminating one variable.
  3. Rozvadov the resulting equation.
  4. Substitute the found value of the variable in any of the original equations.

Examples of solving systems of equations

Solution by graphical methods

Example 1

Rozwarte equation:


Building graphics

Building graphics will see that the graphs intersect at the point


The decision of the method of substitution

Example 2

Rozwarte equation:


From the first equation we Express And substitute the resulting expression into the second equation of the system:

The resulting value is substituted into the expression


Solution by adding

Example 3

Rozwarte equation:


Need to get rid of a variable poslano Multiply the first equation of system 3 and the second by 2.

Add pocino equation and get:

Find the value of the first equation of the system:


Remark: the method add can be multiplied not only on positive numbers, and negative.

How rozwiazywanie the system of equations to solve only to You.

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