document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=75071>Question 75071</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help . And thanks in advance <BR>\n" );
document.write( "Write a system of two linear equations that has .<BR>\n" );
document.write( "a) only one solution ,(2,3).<BR>\n" );
document.write( "b) an infinite number of solutions . <BR>\n" );
document.write( "c) no solution. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #53917 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=53917\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Write a system of two linear equations that has:<BR>\n" );
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document.write( "a) only one solution ,(2,3)<BR>\n" );
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document.write( "An easy way to do this one is to use the slope intercept form y = mx + b. where m is the slope<BR>\n" );
document.write( "of the graph and b is the value where the graph crosses the y-axis.\r<BR>\n" );
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document.write( "For the first linear equation let's say the slope is -2.  Then it's slope intercept form <BR>\n" );
document.write( "becomes y = -2x + b.  We can now solve for b because we know that the point (2,3) is on the<BR>\n" );
document.write( "graph.  Substitute 2 for x and 3 for y and the equation becomes:<BR>\n" );
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document.write( "3 = -2(2) + b<BR>\n" );
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document.write( "multiply on the right side and the equation then becomes:<BR>\n" );
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document.write( "3 = -4 + b<BR>\n" );
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document.write( "Solve for b by adding +4 to both sides to get rid of the -4 on the right side.  When you <BR>\n" );
document.write( "add + 4 to both sides the value of b computes to be:<BR>\n" );
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document.write( "7 = b<BR>\n" );
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document.write( "Substitute this value for b into the slope intercept form and you get:<BR>\n" );
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document.write( "y = -2x + 7<BR>\n" );
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document.write( "This is the first linear equation, and (2,3) is one of the points on its graph.<BR>\n" );
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document.write( "For the second linear equation assume a different value for the slope than you did last time.<BR>\n" );
document.write( "Last time we let the slope be -2.  Suppose this time we make the slope +2 so that we<BR>\n" );
document.write( "know the graphs will not be parallel lines.  The slope intercept form then becomes:<BR>\n" );
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document.write( "y = 2x + b<BR>\n" );
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document.write( "Again we can find b by substituting 3 for y and 2 for x. This leads to:<BR>\n" );
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document.write( "3 = 2(2) + b<BR>\n" );
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document.write( "Multiply out the right side to get:<BR>\n" );
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document.write( "3 = 4 + b<BR>\n" );
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document.write( "Solve for b by subtracting 4 from both sides to get:<BR>\n" );
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document.write( "-1 = b<BR>\n" );
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document.write( "Substitute this value for b into the slope intercept form we are working on, and you get:<BR>\n" );
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document.write( "y = 2x - 1<BR>\n" );
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document.write( "(2, 3) is a point on this graph.  So now we have two equations:<BR>\n" );
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document.write( "y = -2x + 7 and<BR>\n" );
document.write( "y = +2x - 1<BR>\n" );
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document.write( "And both have the common solution of (2,3).<BR>\n" );
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document.write( "b) an infinite number of solutions \r<BR>\n" );
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document.write( "All that you need to do here is to write an equation and make a second equation a multiple<BR>\n" );
document.write( "of the first equation.  Example:<BR>\n" );
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document.write( "The first equation is y = 2x -1<BR>\n" );
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document.write( "The second equation is 2y = 4x - 2<BR>\n" );
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document.write( "This second equation is just 2 times the first equation.  That makes them the same equation<BR>\n" );
document.write( "so they will have the same graph and every solution of the first equation also satisfies the<BR>\n" );
document.write( "second equation.  Therefore, these two equations have an infinite number of common <BR>\n" );
document.write( "solutions.<BR>\n" );
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document.write( "c) no solution.<BR>\n" );
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document.write( "No solution will happen if the two graphs are parallel, but are a separate line.  To make them<BR>\n" );
document.write( "parallel we need only make sure that the linear graphs have the same slope.  However, they<BR>\n" );
document.write( "need to be separate lines, so we can make them cross the y-axis at different points.<BR>\n" );
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document.write( "Let's again use the slope intercept form.  And let's make the slope of both of our lines +3.<BR>\n" );
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document.write( "When we do, the equation becomes y = 3x + b<BR>\n" );
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document.write( "Now we can just substitute two different values for b, the point where the graph crosses the<BR>\n" );
document.write( "y-axis. That forces the graphs to be separate lines.  So we could write:<BR>\n" );
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document.write( "y = 3x + 5 and<BR>\n" );
document.write( "y = 3x - 2<BR>\n" );
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document.write( "These two graphs are parallel because they have the same slope of +3.  Remember that parallel <BR>\n" );
document.write( "lines never cross and to have a common solution, two linear equations must have graphs that <BR>\n" );
document.write( "cross at a point, and the x and y values at that point are the common solution for both<BR>\n" );
document.write( "equations.  So we've made the equations have the same slope +3 making their graphs parallel<BR>\n" );
document.write( "lines.  However, the first equation has a graph that crosses the y-axis at + 5 and the second<BR>\n" );
document.write( "has a graph that crosses the y-axis at -2.  Therefore, the graphs of these two equations<BR>\n" );
document.write( "are parallel but are separated by 7 vertical units (the algebraic difference between 5 and -2).<BR>\n" );
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document.write( "Hope this helps you to see the meaning of 1 common solution, an infinite number of <BR>\n" );
document.write( "solutions, and no common solution and how they tie into the form of the equation and<BR>\n" );
document.write( "the graphs of the two equations. </SPAN>\n" );
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