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Algebra Foundations

Systems of two linear equations are solved using the elimination method.

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After completing this tutorial, you will be able to complete the following:

- Solve systems of linear equations by elimination.
- Determine if a given point is the solution to a linear system.

A linear equation describes a relationship of constant change between variables. A linear equation in two variables, which represents a line in the plane, has the general form . In this notation, x and y represent unknown quantities, or variables; a and b, which represent nonvariable real numbers, are called the coefficients of x and y, respectively. The value of c is also a nonvariable real number.

A linear equation can involve any number of variables, in which case it represents a higher dimensional linear object, such as a plane in the case of three variables, or a hyperplane in four or more variables. The general form of a linear equation in n variables is , where are variables and the coefficients, , are fixed real numbers.

A system of linear equations is a collection of multivariable linear equations with the same variables. For example, the following is a system of linear equations in two variables:

A system of linear equations in two variables can be interpreted as describing lines in the same plane. The following graph shows the lines represented by the system above:

A solution to a system of linear equations is a point whose coordinates, when substituted for the variables, simultaneously satisfy each of the equations. A linear system is consistent if it has a solution. In the example above, is a solution because substituting x = 4 and y = 2 satisfies each equation:

The point is also the intersection point between the lines described by the equations in the system. This fact can provide some insight into linear systems.

Two lines in the plane intersect at exactly one point just in case they not parallel or coincident. Parallel lines do not intersect, whereas coincident lines intersect at infinitely many points because they overlap.

Consequently, a two-variable linear system with two equations can have three types of solution sets. The first type, which corresponds to a system describing skew lines, contains one point-which is the intersection point of the lines. The second and third types are either empty or infinite, corresponding to parallel and coincident lines, respectively. This categorization extends to arbitrary linear systems, so that any linear system has either one solution, no solution, or infinitely many solutions.

A linear system is overdetermined if there are more equations than variables. The solution set to an overdetermined system consists of the points that satisfy all equations in the system. Alternatively, a system is underdetermined if there are fewer equations than variables, and so long as the system is consistent, it is viewed as having infinitely many solutions.

By this reasoning, there are three possibilities for any linear system: exactly one solution, no solutions, or infinitely many solutions.

There are three broadly used methods to find the solution sets of linear systems. The first, which is only efficient for certain two or three variable systems, is to graph the linear surfaces and find their intersection by inspection. The second method, again most useful for systems involving few variables, is called the substitution method. Finally, there is the elimination method for linear systems that are written in standard form.

The elimination method specifies three operations that, when applied to a system, yield a system of equations with the same solution set as the original. Two of these operations allow for any equation in the system to be either multiplied by a nonzero constant, or replaced by its sum with another equation in the system. The third operation involves switching the ordering of equations in the system. Strategically applied, these operations "eliminate" variables from the system by solving for their values.

We can use the previously discussed linear system to demonstrate this method:

To solve this system by elimination, begin by multiplying both sides of the first equation by ?2. The purpose of this step is best understood in retrospect:

Next, we add the first equation to the second, thereby eliminating x from the second equation:

The second equation becomes a one-variable equation with solution y = 2. This value is substituted into the first equation for y, which yields therefore, x = 4. We see that the elimination method yields the previously determined solution, . Note that the initial step of multiplying the first equation by ?2 provided the conditions for x to beliminated from the second equation. This strategy works for any linear system.

In the case that there are infinitely many solutions to the system, elimination will eventually produce an equation of the form c = c, where c is some number. For example, the following system has infinitely many solutions, which is evidenced by the fact that the second equation is a constant multiple of the first:

In the case of a system with no solutions, which occurs in two variables when the equations describe parallel lines, elimination will eventually produce a line of the form , where c is nonzero. For example, the following system has no solutions, given that the two equations describe parallel noncoincident lines:

By following the same steps as before, we obtain the line , which cannot be satisfied by any x.

In order to successfully apply the elimination method to a given system, we must ensure that each of the equations is written in standard form. Consider, for example, a system of equations written in nonstandard form:

The elimination method cannot successfully be applied to this system. Furthermore, the elimination method cannot be applied to general nonlinear systems, such as a system of polynomial equations. Such possible misapplications of the method are prohibited by definition: the elimination method is defined only for linear systems in standard form.

Approximate Time | 30 Minutes |

Pre-requisite Concepts | Students should know how to evaluate algebraic expressions and solve linear equations. |

Course | Algebra Foundations |

Type of Tutorial | Skills Application |

Key Vocabulary | coefficient, elimination method, linear equation |