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Algebra Foundations

Systems of two linear equations are solved graphically.

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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.

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 ax+by=c. 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. The general form of a linear equation in n variables is whereare 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 satisfies each of the equations. In the example above, (4, 2) is a solution because substituting x = 4 and y = 2 satisfies each equation.

The point (4, 2) 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 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.

Approximate Time | 30 Minutes |

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

Course | Algebra Foundations |

Type of Tutorial | Skills Application |

Key Vocabulary | coefficient, graph linear equation, linear equation |