In a context of this article, system of linear equations refers to the collection of two simultaneous equations with two unknown variables x and y, expressed in canonical form as:
a1*y + b1*y = c1
a2*x + b2*y = c2
Following is the numeric example, corresponding to the symbolic notation shown above:
2x + 5y = 31
3x + 4y = 29
A solution to the system of two linear equations with two unknown variables x and y is defined as an ordered pair of numbers, which could be assigned to the variables x and y to satisfy both equations. In our numeric example such solution is: x= 3; y=5.
To test the solution, plug the values for x and y in each equation and check its validity. In our example:
2*3 + 5*5 = 31 (correct)
3*3 + 4*5 = 29 (correct)
There are three possible cases regarding the solution to the system of two linear equations with two unknown variables:
1. No solution
2. Exactly one solution
3. Infinite number of solutions
System said to be consistent if it has at least one solution; otherwise, it’s called inconsistent. The equations, which comprise the system, called independent if they don’t’ share all solutions. Independent equations should have exactly one common solution; otherwise the equations are dependent. Typically, dependent equations could be transformed to each other by simple multiplication. In this regards, the following equations are dependent because the second one is just a product of the first one and the integer multiplier of 2 (the multiplier actually could be any rational number, still resulting in the pair of dependent equations):
2 x + 3 y = 5
4x + 6y = 10
Dependent equations have infinite number of solutions, and typically are not of much interest. A system of two linear equations with two unknown variables represent the simplest case (a sub-class) of the linear simultaneous equations class, which in general form contains N equations with N unknown variables. The necessary (not sufficient!) condition for the system of linear equations to have a unique solution is: the number of variables should be equal to the number of equations. Please note, that this condition is not sufficient and does not guarantee the unique solution; rather from the opposite, it states that if the condition is not met, then the system will have either infinite number of solutions, or no solution at all.
System of linear equations could be solved in many different ways. Various algorithms exist and among them the most popular and widely used ones are:
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Elimination of variables
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Substitution method, conceptually similar to the elimination of variables
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Cramer’s rule
Elimination method serves as a conceptual umbrella, covering the set of a essentially similar mathematical algorithms, known as addition method and more general, Gaussian elimination method. The core idea behind these methods is to transform the system of linear equations into another equivalent system, where each equation contains only one unknown variable, and thus could be easily solved. Following are the symbolic and numeric examples:
2x + 7y = 31
3x + 4y = 27
Multiplying the first equation by (-3/2) will result in another equivalent system of equations:
-3x - 21/2* y = -93/2
3x + 4*y = 27
Adding these two equations will result in equation with only one variable y:
4*y – 21/2 * y = 27 – 93/2
Correspondingly, the solution for y is:
y= (- 39/2) / (-13/2) = 3
By plug-in the numeric value for y=3 in any of the original equations (for certainty, in the first one) the solution for variable x could be found as shown below:
2X + 21 = 31, and correspondingly: X = 5
Substitution method, which falls under conceptual umbrella of the elimination methods, is explained below in symbolic form as a sequence of steps:
From the first symbolic equation derive the expression for x = ( c1 –b1*y ) / a1. Substitute the variable x in the second equation, resulting in a linear equation with single variable y:
a2*( c1 –b1*y ) / a1 + b2 * y = c2
Simplify the latter one :
(a2 * ( c1 –b1*y ) / a1 + b2 ) * y= c2
a2*(c1-b1*y) + a1*b2* y = a1*c2
y*(a1*b2-a2*b1)=a1*c2-a2*c1
And finally, solve the equation for variable y = c2 / (a2 * (c1 –b1*y ) / a1 + b2 )
Cramer’s rule provides the most straightforward symbolic solution to the system of two linear equations based on the fundamental concept of determinant, defined as: det = a1*b2 – a2*b1. The solution for each variable could be found just in a single step as shown below:
x = (b2 * c1 - b1 * c2 ) / det
y = (a1 * c2 - a2 * c1 ) / det
Wrapping up the abstract part and adding a brief note on comparative advantages of the methods discussed above. Cramer’s rule is very computationally efficient for the “small matrices”, in other words, for relatively small system of linear equations, contain just 2, 3, 4 equations with corresponding number of unknown variables. For the much bigger matrices, other methods provide better computation efficiency, though the detailed discussion is going far beyond the boundary of a single article.
Couple words about geniuses of the past, who greatly contributed to the solutions:
Johann Carl Friedrich Gauss (1777-1855), German mathematician and scientist, widely recognized as one of the greatest of all times.
Gabriel Cramer (1704-1752) published the now famous Cramer’s rule in 1750 and, thus, successfully secured his place in the eternal hall of math fame.
In order to solve the system of linear equations online follow the steps:
Open web browser and navigate it to the free online system equation solver. You should see a web page looking similar to the sample snapshot, shown on the image attached to the article.
Enter the coefficients of the both linear equations and click on the “solve” screen button. The application will try to find the solution and upon success return the ordered pair of number, corresponding to variables x and y.
The users of portable Internet-enabled devices, like Apple iPod Touch™, iPhone™, Microsoft Zune™ HD, SmartPhones™) should navigate the browser to the mobile system equation solver , optimized for the mobile devices with reduced screen.
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