October 15, 2008

This resource has been contributed by Winpossible, and can also be accessed on their website by clicking here - Determine if a system of equations has infinite solutions.

In this mini-lesson, you will learn how to determine, if a system of equations has infinite solutions. A system of equations has infinite solutions when the lines are parallel, i.e. they have the same slope, and they have the same y-intercept. In fact one equation is a scalar multiple of the other and hence, in effect, the equations represent the same line! Let us look at system of two linear equations Ax + By + C = 0 and Dx + Ey + F = 0: these equations will have infinite solutions if the ratio of A/D, B/E and C/F are the same i.e. A/D = B/E = C/F. In such a case, these lines represent coincident lines, i.e. they overlap at every single point. For example, x + y = 2 and 3x + 3y = 6 have infinite solutions because A/D = B/E = C/F = 1/3. Another way to look at this is: if you multiply line 1 by three you get line 2, and thus these two lines are exactly the same line!

In this mini-lesson, you will learn how to determine, if a system of equations has infinite solutions. A system of equations has infinite solutions when the lines are parallel, i.e. they have the same slope, and they have the same y-intercept. In fact one equation is a scalar multiple of the other and hence, in effect, the equations represent the same line! Let us look at system of two linear equations Ax + By + C = 0 and Dx + Ey + F = 0: these equations will have infinite solutions if the ratio of A/D, B/E and C/F are the same i.e. A/D = B/E = C/F. In such a case, these lines represent coincident lines, i.e. they overlap at every single point. For example, x + y = 2 and 3x + 3y = 6 have infinite solutions because A/D = B/E = C/F = 1/3. Another way to look at this is: if you multiply line 1 by three you get line 2, and thus these two lines are exactly the same line!

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Algebra I. Click on the video below to go through it. If you like it, you can buy our online course in Algebra I by clicking here.

- Mathematics > General
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Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Use appropriate tools strategically.

Attend to precision.

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.