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Solving Systems of Linear Equations Graphically

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Systems of Linear Equations and Inequalities

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Solving Systems of Linear Equations Graphically

Algebra-1

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You will solve systems of two linear equations graphically.

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Now You Know

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

  • Solve systems of linear equations graphically.
  • Determine if a given point is the solution to a linear system.

Everything You'll Have Covered

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:

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

Consequently, a two-variable system of linear equations can have three types of solution sets or simultaneous solutions to the system. The first type of solution set is that which contains a single point corresponding to the intersection point of skew lines. The second and third types are either infinite or empty, 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.

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; this method is sometimes called the graphical method.

In order to solve a linear system graphically, one should write each equation in slope-intercept form and then graph them on the same axes. Care should be taken to choose an appropriate scale, which could require algebraic reasoning or guess-and-check methods.

Tutorial Details

Approximate Time 30 Minutes
Pre-requisite Concepts Students should know how to evaluate algebraic expressions and graph linear equations.
Course Algebra-1
Type of Tutorial Skills Application
Key Vocabulary graph linear equation, linear equation, solution of a system of linear equations