Unit+3+-+System+of+equations

In [|mathematics], simultaneous equations are a set of [|equations] containing multiple variables. This set is often referred to as a system of equations. A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations. To find a solution, the solver needs to use the provided equations to find the exact value of each variable. Generally, the solver uses either a [|graphical method], the [|matrix] method, the substitution method, or the elimination method. Some textbooks refer to the elimination method as the addition method, since it involves adding equations (or constant multiples of the said equations) to one another, as detailed later in this article.

This is a set of [|linear equations], also known as a [|linear system of equations] : Finding Solutions: Solving this involves subtracting x + y = 6 from 2x + y = 8 (using the elimination method) to remove they-variable, then simplifying the resulting equation to find the value of x, then substituting the x-value into either equation to find y. The solution of this system is: which can also be written as an [|ordered pair] (2, 4), representing on a graph the coordinates of the point of intersection of the two lines represented by the equations.

Sometimes not all variables can be solved for, and so an answer for at least one variable must be expressed in terms of other variables and so the set of all solutions is infinite; this is typical for the case where the system has fewer equations than variables. If the number of equations is the same as the number of variables, then probably (but not necessarily) the system is exactly solvable in the sense that the set of its solutions is finite; for a [|system of linear equations] in this case there is exactly one solution, for other systems to have several solutions is also typical. Sometimes a system has no solution; this is typical for the case where the system has more equations than variables. If these rules about connection between number of solutions and numbers of equations and variables do not hold, then such situation is often referred to as dependence between equations or between their left parts. For instance, this occurs in linear systems if one equation is a simple multiple of the other (representing the same line, e.g. 2x + y = 3 and 4x + 2y = 6) or if the ratio of like variables in two linear equations is the same (representing parallel lines, e.g. 2x + y = 3 and 6x + 3y = 7 where the ratio of comparable letters is 3). Systems of two equations in two real-value unknowns usually appear as one of five different types, having a relationship to the number of solutions: The equation x2 + y2
 * 1) Systems that represent intersecting sets of points such as lines and curves, and that are not of one of the types below. This can be considered the normal type, the others being exceptional in some respect. These systems usually have a finite number of solutions, each formed by the coordinates of one point of intersection.
 * 2) Systems that simplify down to false (for example, equations such as 1 = 0). Such systems have no points of intersection and no solutions. This type is found, for example, when the equations represent parallel lines.
 * 3) Systems in which both equations simplify down to an identity (for example, x = 2x − x and 0y = 0). Any assignment of values to the unknown variables satisfies the equations. Thus, there are an infinite number of solutions: all points of the plane.
 * 4) Systems in which the two equations represent the same set of points: they are mathematically equivalent (one equation can typically be transformed into the other through algebraic manipulation). Such systems represent completely overlapping lines, or curves, etc. One of the two equations is redundant and can be discarded. Each point of the set of points corresponds to a solution. Usually, this means there are an infinite number of solutions.
 * 5) Systems in which one (and only one) of the two equations simplifies down to an identity. It is therefore redundant, and can be discarded, as per the previous type. Each point of the set of points represented by the other equation is a solution of which there are then usually an infinite number.

[[|edit]]Substitution Method
The two example [|equations] intersect twice. Therefore, there are two solutions.

Systems of simultaneous equations can be hard to solve unless a systematic approach is used. A common technique is the substitution method: Find an equation that can be written with a single variable as the subject, in which the left-hand side variable does not occur in the right-hand side expression. Next, [|substitute] that expression where that variable appears in the other equations, thereby obtaining a smaller system with fewer variables. After that smaller system has been solved (whether by further application of the substitution method or by other methods), substitute the solutions found for the variables in the above right-hand side expression. In this set of equations x is made the subject of the second equation: then, this result is substituted into the first equation: After simplification, this yields the solutions and by substituting this in x = −2y the corresponding x values are obtained. The two solutions of the system of equations are then:

Substitution Method Steps!

 * 1) Arrange equation so that one variable is by itself.
 * 2) Substitute the new equation into the other original equation.
 * 3) Solve.
 * 4) Substitute answer into either of the original equations.
 * 5) Solve for other variable.
 * 6) Write answer as an ordered pair (X,Y)

Elimination Method
Elimination by judicious multiplication is the other commonly used method to solve simultaneous linear equations. It uses the general principles that each side of an equation still equals the other when both sides are multiplied (or divided) by the same quantity, or when the same quantity is added (or subtracted) from both sides. As the equations grow simpler through the elimination of some variables, a variable will eventually appear in fully solvable form, and this value can then be "back-substituted" into previously derived equations by plugging this value in for the variable. Typically, each "back-substitution" can then allow another variable in the system to be solved.

Elimination Method Steps!

 * 1) Arrange equation with the like terms in columns.
 * 2) Multiply one or both equations by a number to get opposites for one of the variables (X or Y).
 * 3) Add like terms to eliminate said variable.
 * 4) Solve for the remaining variable.
 * 5) Substitute the variable you just sloved for into one of the original equations.
 * 6) Solve for the other variable.

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media type="youtube" key="XQ7aNAxZ0s0?fs=1" height="385" width="480" Video on System of Equations

= Websites!! = __[]__ This site was helpful in learning the elimination because it gave some practice problems and answers along with definitions and other helpful tools to use.

[] This was helpful in learning the substitution methood, it gave step by step directions. And best of all, it used easy to understand language instead of big words that no one understands.

[] This website show more information on the graphing method. There was explinations and a "mathematical definition" along with a video showing how to solve some problems.

[] There are examples and step by step directions on how to use the graphing calculator method. There were pictures of what you calculator screen should look like to help also.

[] This site had a video as well as step by step directions for substitution, elimination and the graphing method.