document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203091>Question 1203091</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For the topic Linear equations inequalities create two unknowns (for example 10x+4y=72 and 4x+8y=48) but the intercept must be x=1 and y=19 or (1,19) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838366 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=838366\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Any linear equation in standard form is Ax+By = C<BR>\n" );
document.write( "A,B,C are integers.\r<BR>\n" );
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document.write( "For example: 2x+3y = 4\r<BR>\n" );
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document.write( "If we have the 2x+3y portion, then plugging in (x,y) = (1,19) leads to:<BR>\n" );
document.write( "2x+3y <BR>\n" );
document.write( "2*1+3*19 <BR>\n" );
document.write( "2+57<BR>\n" );
document.write( "59\r<BR>\n" );
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document.write( "This means 2x+3y = 59 when (x,y) = (1,19) <BR>\n" );
document.write( "Rephrased another way: (x,y) = (1,19) is a solution to 2x+3y = 59.<BR>\n" );
document.write( "Another rephrasing: The point (1,19) is on the line 2x+3y = 59\r<BR>\n" );
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document.write( "Now let's generate the other equation.<BR>\n" );
document.write( "Pick another two random integers.<BR>\n" );
document.write( "Let's say we go for 5 and 6 to generate 5x+6y<BR>\n" );
document.write( "Plug x = 1 and y = 19 into that expression<BR>\n" );
document.write( "5x+6y<BR>\n" );
document.write( "5*1+6*19<BR>\n" );
document.write( "5+114<BR>\n" );
document.write( "119\r<BR>\n" );
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document.write( "Therefore, 5x+6y = 119 when x = 1 and y = 19<BR>\n" );
document.write( "In other words, the ordered pair (x,y) = (1,19) is a solution to 5x+6y = 119<BR>\n" );
document.write( "The point (1,19) is on the line 5x+6y = 119.\r<BR>\n" );
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document.write( "We've demonstrated that (x,y) = (1,19) is a solution to both 2x+3y = 59 and 5x+6y = 119 simultaneously.<BR>\n" );
document.write( "This point is on both lines at the same time.<BR>\n" );
document.write( "This means the two lines intersect at (1,19).<BR>\n" );
document.write( "Use Desmos, GeoGebra, TI83/84, or any graphing tool to visually confirm this.\r<BR>\n" );
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document.write( "A non-visual way to confirm is to use any of the following:<ul><li>Substitution</li><li>Elimination</li><li>Matrix Row Reduction (either REF or RREF)</li><li>Cramer's Rule</li></ul>As the tutor Theo has shown using Desmos, there is another system of equations that yields the solution (x,y) = (1,19)\r<BR>\n" );
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document.write( "It turns out there are infinitely many systems possible. I encourage you to explore another possible system.<BR>\n" );
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