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Solving Systems of Linear Equations Using the Elimination Method

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Solving Systems of Linear Equations Using the Elimination Method


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You will solve systems of linear equations using the elimination method.

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Now You Know

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.

Everything You'll Have Covered

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.

Tutorial Details

Approximate Time 30 Minutes
Pre-requisite Concepts Students should understand how to evaluate algebraic expressions and solve linear equations.
Course Algebra-1
Type of Tutorial Procedural Development
Key Vocabulary elimination method, linear equation, solution of a system of linear equations