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Translations of Encyclopedia about Mathematics

 

Linear Equations

The following passages will cover those special cases when the number of unknown variables equals the number of given equations, a situation which has its foundation and particular significance when used practically.

Two Linear Equations with Two Unknown Variables / Concept of the System of Equations

A linear equation with two unknown variables can always be converted to the normalised form ax1+bx2=c (where x1 a x2 are unknowns and a, b, and c are some real constants). Every unknown value of x1 can be assigned a certain unknown value of x2, and vice versa.

Example: 5 x1 4x2 = 16. For this equation there exists an infinite number of numerical pairs which would satisfy it. If a second equation is given which would be independent of the first and which does not contradict it (by independent we mean that the second equation cannot be derived from a transformation of the first; by not contradicting it we mean that one equation does not exclude the second, as would be the case if the set of two equations were the following: 5 x1 4 x2 = 16 a 5 x1 4 x2 = 17), then there exists only one pair of numbers which would fulfil the conditions of both equations. The equations a1x1 + b1x2 = c1 and a2x1 + b2x2 = c2 creates a system of linear equation with two unknown variables (a system of equations). The constants a1, a2, b1, b2, c1 and c2 are real numbers.

Solving a System of Equations

The ordered pairs (x1; x2) which satisfy both equations are called the solution for the system of equations. We can solve this problem either by calculation or with the use of graphs. A test involves attaining the correct results ( x1 ; x2) in the system of equations.

Solving by Calculation Elimination Method

The principle of this method relies on deriving a new equation from those given, where the new equation will contain only one of the unknowns and where the other is eliminated. This process, which is referred to as the elimination method, can be achieved in several manners, where which one of the unknowns will be eliminated or what approach will be used depends on the structure of the given equations.

There are three methods how to proceed with the elimination method:

  1. The addition method, which uses the same coefficients, is based on the following principle: if this type of equation is multiplied by a certain number such that the coefficients of one of the unknowns in both equations was a negation of the other (as in a1 = -a2), then, in the second step, which is addition, the unknown will drop out. This addition method is used when the coefficients of one of the unknowns are equal or when the equation can be easily transformed such that they would be equal.

  1. The substitution method is based on the following principle: one of the unknowns in one of the two equations is isolated and the attained equation is inserted into the second equation in place of that unknown.

  2. The comparative method is based on the following principle: the same unknown is isolated in both of the equations and the attained expressions are combined into a single equation. The comparative method is used when one unknown can be easily isolated in both of the equations.

 

Actual Calculation - Working Procedure

1 Consider both systems of equations as analytical expressions of functions. The graphical representation of these functions within the x, y coordinate system are bisectors.
2 The coordinate of intersection points can be cancelled from the graph and represents a solution to the system of equations.
3 The graphical method of solution (approximate solution) must always be tested using the original equations.

Linear Equations with More than Two Unknowns

Statements which are valid in a system of equations with two unknowns can be transformed to systems with more unknowns. A standard system of equations with three unknowns can look like:

a1x1 + b1x2 + c1x3 = d1
a2x1 + b2x2 + c2x3 = d2
a3x1 + b3x2 + c3x3 = d3

The three values (x1; x2; x3) which would satisfy all three equation are called the solution to the system of equations. We can determine exactly one solution if all of the equations are independent of and do not contradict one another. If two or three equations would be dependent on one another, the system of equations will have an unlimited number of solutions. If two or more equations would contradict one another, there would be no solution at all. A test requires substituting the solution to our three values (x1, x2, x3) into the three equations.

Solving linear systems of equations with three unknowns is accomplished in the following manner:

a) eliminating one of the three unknowns allows us to combine the system into only two equations having two unknowns;
b) we then solve the resulting system of two equation using the elimination method;
c) the third and last unknowns are solved by substituting the two solved unknowns into the final equation.

Non-linear Systems of Equations

Besides the linear system of equations, there are also various difficult, non-linear systems of equations, which are systems whose equations contain expressions higher than the first order. There are no general rules for solving these types of equation systems, although graphical representation may help us in this. In this manner, we are able to determine the coordinates of intersection points or points intersecting with curves corresponding to the individual equations. Intersection points or contact points are the real solutions for these systems of equations.

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