document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=250122>Question 250122</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi my teacher wants us to determin what each linear graph is, But I have always had trouble with linear equations I was wondering if you could just send me an example of the three different graphs and an explantion for each (independant, dependent, and inconsistent). So I can try to do it myself. thank you so much!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #182098 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=182098\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> linear equations in two dimensions are equations of lines.\r<BR>\n" );
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document.write( "they have the form ax + by = c where a is the coefficient of the x term and b is the coefficient of the y term and c is a constant.\r<BR>\n" );
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document.write( "the slope intercept form of these equations is y = m*x + b where m is the slope and b is the y-intercept (value of y when x = 0).\r<BR>\n" );
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document.write( "when you graph these equations, you use the slope-intercept form.\r<BR>\n" );
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document.write( "to convert from the standard form of the equation to the slope-intercept form of the equation, you solve for y.\r<BR>\n" );
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document.write( "no solution occurs when the lines do not intercept.  this means that they are parallel to each other.\r<BR>\n" );
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document.write( "here's a graph of 2 linear equations where there is no solution.\r<BR>\n" );
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document.write( "the equations for these lines are:\r<BR>\n" );
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document.write( "y = x+3<BR>\n" );
document.write( "y = x-3\r<BR>\n" );
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document.write( "not the slopes are equal.\r<BR>\n" );
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document.write( "since the lines are parallel, they will never intersect.\r<BR>\n" );
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document.write( "independent solution occurs when the lines do intersect.   this means that there is one unique solution to the problem.\r<BR>\n" );
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document.write( "here's a graph of 2 linear equations where the lines intersect.\r<BR>\n" );
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document.write( "the equations for these lines are:\r<BR>\n" );
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document.write( "y = x+3<BR>\n" );
document.write( "y = -x-3\r<BR>\n" );
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document.write( "note that the slopes are not equal.\r<BR>\n" );
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document.write( "these lines are not parallel and they are not identical so they will always intersect.\r<BR>\n" );
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document.write( "dependent solutions occur when the lines are identical.\r<BR>\n" );
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document.write( "there are multiple solutions to these equations.\r<BR>\n" );
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document.write( "each equation is usually identical to, or an exact multiple of, the other.\r<BR>\n" );
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document.write( "these lines have multiple solutions because they are the same line.\r<BR>\n" );
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document.write( "an example of 2 such equations in standard form would be:\r<BR>\n" );
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document.write( "5x + 3y = 7<BR>\n" );
document.write( "10x + 6y = 14\r<BR>\n" );
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document.write( "solve for y in both equations and you will get an identical equation.\r<BR>\n" );
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document.write( "the first equation becomes y = -5x/3 + 7/3\r<BR>\n" );
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document.write( "the second equation becomes y = -10x/6 + 14/6\r<BR>\n" );
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document.write( "since 10/6 is equivalent to 5/3 and 14/6 is equivalent to 7/3, the second equation simplifies to y = -5x/3 + 7/3\r<BR>\n" );
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document.write( "these lines will have multiple solutions because they are the same line so the lines will intersect at all points.\r<BR>\n" );
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document.write( "here's a graph of the identical lines.\r<BR>\n" );
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document.write( "it looks like one line but it is really 2 identical lines superimposed on each other.\r<BR>\n" );
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document.write( "slope intercept form of the no solution equations are:\r<BR>\n" );
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document.write( "y = x-3<BR>\n" );
document.write( "y = x+3\r<BR>\n" );
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document.write( "standard form of these equations is:\r<BR>\n" );
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document.write( "-x + y = -3<BR>\n" );
document.write( "-x + y = 3\r<BR>\n" );
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document.write( "subtract second equation from first equation and you get:\r<BR>\n" );
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document.write( "0 + 0 = -6 which becomes 0 = -6 which is NOT true therefore these is no solution.\r<BR>\n" );
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document.write( "slope intercept form of the independent solution equations is:\r<BR>\n" );
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document.write( "y = x+3<BR>\n" );
document.write( "y = -x-3\r<BR>\n" );
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document.write( "standard form of these equations is:\r<BR>\n" );
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document.write( "-x + y = 3<BR>\n" );
document.write( "x + y = -3\r<BR>\n" );
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document.write( "subtract second equation from first equation to get:\r<BR>\n" );
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document.write( "-2x = 6\r<BR>\n" );
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document.write( "solve for x to get x = -3\r<BR>\n" );
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document.write( "solve for y to get y = 0\r<BR>\n" );
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document.write( "there is a unique solution at (x,y) = (-3,0) because the lines intersect at that point and 2 lines in the same plane can intersect in one and only one point.\r<BR>\n" );
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document.write( "standard form of the dependent solution equations is:\r<BR>\n" );
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document.write( "5x + 3y = 7<BR>\n" );
document.write( "10x + 6y = 14\r<BR>\n" );
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document.write( "multiply first equation by 2 to get:\r<BR>\n" );
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document.write( "10x + 6y = 14<BR>\n" );
document.write( "10x + 6y = 14\r<BR>\n" );
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document.write( "subtract second equation from first equation to get:\r<BR>\n" );
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document.write( "0 + 0 = 0 which becomes 0 = 0\r<BR>\n" );
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document.write( "the variable dropped out and the equation IS true, so there are multiple solutions to this problems.\r<BR>\n" );
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document.write( "any value of x will lead to a value of y that automatically applies to both equations because these equations are identical.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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