document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196725>Question 1196725</A>: <I><SPAN STYLE=\"word-break: break-all;\"> -4x+9y=9<BR>\n" );
document.write( "x-3y=-6<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829694 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=829694\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll go over 3 ways to solve.\r<BR>\n" );
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document.write( "Method 1) Elimination\r<BR>\n" );
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document.write( "We have 9y in the first equation and -3y in the second.<BR>\n" );
document.write( "If we can get the coefficients to be equal and opposite, then we can add them to cancel out.\r<BR>\n" );
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document.write( "So let's triple everything in the second equation to go from x-3y = -6 to 3x-9y = -18\r<BR>\n" );
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document.write( "We have this equivalent system<BR>\n" );
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document.write( "Add the equations straight down<BR>\n" );
document.write( "The x terms add to -4x+3x = -x<BR>\n" );
document.write( "The y terms cancel out 9y+(-9y) = 0y = 0<BR>\n" );
document.write( "The right hand sides add to -9 since 9+(-18) = -9\r<BR>\n" );
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document.write( "We're left with -x = -9 which solves to x = 9\r<BR>\n" );
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document.write( "Then plug this into any previous equation involving x and y<BR>\n" );
document.write( "-4x+9y = 9<BR>\n" );
document.write( "-4(9)+9y = 9<BR>\n" );
document.write( "-36+9y = 9<BR>\n" );
document.write( "9y = 9+36<BR>\n" );
document.write( "9y = 45<BR>\n" );
document.write( "y = 45/9<BR>\n" );
document.write( "y = 5\r<BR>\n" );
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document.write( "The solution is (x,y) = (9,5)\r<BR>\n" );
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document.write( "Method 2) Substitution\r<BR>\n" );
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document.write( "With this method, we look for variables where the coefficients are easy to work with. In this case, 1 is easiest since dividing by it doesn't change the number.\r<BR>\n" );
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document.write( "The equation x-3y = -6 is the same as 1x-3y = -6\r<BR>\n" );
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document.write( "Let's solve for x in this equation<BR>\n" );
document.write( "x-3y = -6<BR>\n" );
document.write( "x = -6+3y<BR>\n" );
document.write( "Then plug this into the other equation and solve for y\r<BR>\n" );
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document.write( "-4x+9y = 9<BR>\n" );
document.write( "-4(-6+3y)+9y = 9<BR>\n" );
document.write( "24-12y+9y = 9<BR>\n" );
document.write( "24-3y = 9<BR>\n" );
document.write( "-3y = 9-24<BR>\n" );
document.write( "-3y = -15<BR>\n" );
document.write( "y = -15/(-3)<BR>\n" );
document.write( "y = 5\r<BR>\n" );
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document.write( "Now use this to find x<BR>\n" );
document.write( "x = -6+3y<BR>\n" );
document.write( "x = -6+3*5<BR>\n" );
document.write( "x = -6+15<BR>\n" );
document.write( "x = 9\r<BR>\n" );
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document.write( "We arrive at the same solution (x,y) = (9,5)\r<BR>\n" );
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document.write( "Method 3) Graphing\r<BR>\n" );
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document.write( "I'll use technology to graph each equation. Though you could graph each line by hand. The idea behind that is to plug in various x values to find corresponding paired y values. Then draw a straight line through each (x,y) point on the graph.\r<BR>\n" );
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document.write( "Here's the graph if you were to use Desmos<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.desmos.com/calculator/pipioay22g\">https://www.desmos.com/calculator/pipioay22g</a><BR>\n" );
document.write( "Though you can use whichever app you prefer better. GeoGebra is another good option.\r<BR>\n" );
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document.write( "The two lines intersect at (x,y) = (9,5)<BR>\n" );
document.write( "Graphing is a great way to visually confirm the answers quickly. <BR>\n" );
document.write( "However, a more formal way to confirm the answers is detailed in the next section below.\r<BR>\n" );
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document.write( "Check:<BR>\n" );
document.write( "Plug x = 9 and y = 5 into the first original equation<BR>\n" );
document.write( "-4x+9y = 9<BR>\n" );
document.write( "-4*9+9*5 = 9<BR>\n" );
document.write( "-36+45 = 9<BR>\n" );
document.write( "9 = 9<BR>\n" );
document.write( "Since we arrive at a true statement, this makes the original equation true when we plug in those x,y values mentioned.\r<BR>\n" );
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document.write( "Repeat similar steps for the second original equation<BR>\n" );
document.write( "x-3y = -6<BR>\n" );
document.write( "9-3*5 = -6<BR>\n" );
document.write( "9-15 = -6<BR>\n" );
document.write( "-6 = -6<BR>\n" );
document.write( "Both equations are true for these x,y values. <BR>\n" );
document.write( "The answer is confirmed.\r<BR>\n" );
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document.write( "This system is consistent and independent since we get one single solution. <BR>\n" );
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