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solution of linear Equations: Graphing(page 2of 7)

You are watching: If lines are parallel how many solutions   When you are resolving systems the equations (linear or otherwise), girlfriend are, in terms of the equations" related graphed lines, finding any intersection points of those lines.

For two-variable linear systems of equations, there are then three possible types of remedies to the systems, which correspond to three different species of graphs the two directly lines.

These three situations are depicted below:

 Case 1 Case 2 Case 3

The an initial graph above, "Case 1", mirrors two unique non-parallel lines the cross at precisely one point. This is dubbed an "independent" system of equations, and the equipment is constantly some x,y-point.

 Independent system: one systems point Case 2 Case 3

The second graph above, "Case 2", reflects two distinctive lines that space parallel. Because parallel lines never ever cross, climate there have the right to be no intersection; the is, because that a mechanism of equations that graphs as parallel lines, there can be no solution. This is referred to as an "inconsistent" mechanism of equations, and also it has no solution.

 Independent system: one solution and one intersection point Inconsistent system: no solution and no intersection point Case 3

The third graph above, "Case 3", appears to show only one line. Actually, it"s the exact same line drawn twice. These "two" lines, really being the same line, "intersect" in ~ every point along your length. This is referred to as a "dependent" system, and also the "solution" is the entirety line.

 Independent system: one solution and also one intersection point Inconsistent system: no solution and also no intersection point Dependent system: the solution is the totality line

This shows that a mechanism of equations may have one systems (a certain x,y-point), no systems at all, or an unlimited solution (being all the solutions to the equation). You will never have a system with two or 3 solutions; it will constantly be one, none, or infinitely-many.

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 because that instance, if the lines cross at a shallow edge it deserve to be just about impossible come tell where the currently cross. And if the intersection point isn"t a succinct pair of whole numbers, every bets space off. (Can friend tell by looking that the displayed solutionhas works with the (–4.3, –0.95)? No? then you watch my point.) On the plus side, because they will certainly be forced to provide you pretty neat options for "solving by graphing" problems, girlfriend will be able to get every the right answers as lengthy as friend graph an extremely neatly. For instance:

deal with the following system by graphing.

2x – 3y = –2 4x + y = 24

I recognize I need a practiced graph, for this reason I"ll grab my ruler and get started. First, I"ll deal with each equation because that "y=", so I deserve to graph easily:

2x – 3y = –2 2x + 2 = 3y (2/3)x + (2/3) = y

4x + y = 24 y = –4x + 24

The 2nd line will be easy to graph using simply the slope and also intercept, but I"ll require a T-chart for the very first line.

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 x y = (2/3)x + (2/3) y = –4x + 24 –4 –8/3 + 2/3 = –6/3 = –2 16 + 24 = 40 –1 –2/3 + 2/3 = 0 4 + 24 = 28 2 4/3 + 2/3 = 6/3 = 2 –8 + 24 = 16 5 10/3 + 2/3 = 12/3 = 4 –20 + 24 = 4 8 16/3 + 2/3 = 18/3 = 6 –32 + 24 = –8

(Sometimes you"ll an alert the intersection right on the T-chart. Carry out you check out the point that is in both equationsabove? inspect the gray-shaded row above.)

 now that I have actually some points, I"ll grab mine ruler and graph neatly, and look because that the intersection: Even if ns hadn"t noticed the intersection allude in the T-chart, ns can absolutely see it from the picture. solution: (x, y) = (5, 4)

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