document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=126657>Question 126657</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0072549017/algebrahomeworkh>Beginning Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> 8x-4y=16<BR>\n" );
document.write( "    y=2x-4<BR>\n" );
document.write( "Solve by substition\r<BR>\n" );
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document.write( "4y=8x-16<BR>\n" );
document.write( "divide both sides by 4y<BR>\n" );
document.write( "y=8x-16/4<BR>\n" );
document.write( "y=8x/4+16/4<BR>\n" );
document.write( "y=4x/2+16/4<BR>\n" );
document.write( "reduce<BR>\n" );
document.write( "y=2x+4\r<BR>\n" );
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document.write( "Upon plugging top equation of y into bottom equation, I became confused I came up with <BR>\n" );
document.write( "y=2x-4 and y=2x+4<BR>\n" );
document.write( "I must have done something wrong along the line\r<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 #92805 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=92805\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given the following pair of equations:<BR>\n" );
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document.write( "8x - 4y = 16<BR>\n" );
document.write( "y = 2x - 4<BR>\n" );
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document.write( "To solve a pair of equations, you need to solve one of the equations for one variable in<BR>\n" );
document.write( "terms of the other variable. Then you substitute that equivalent value into the other equation <BR>\n" );
document.write( "and solve it.<BR>\n" );
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document.write( "Notice that the second equation has already been solved for y in terms of x. Therefore,<BR>\n" );
document.write( "you already know that y is equal to 2x - 4.<BR>\n" );
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document.write( "Now go to the first equation and substitute that value (2x - 4) for y in that equation.<BR>\n" );
document.write( "When you do that substitution the first equation becomes:<BR>\n" );
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document.write( "8x - 4(2x - 4) = 16<BR>\n" );
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document.write( "You now have an equation that contains only one variable, so it can be solved to get that <BR>\n" );
document.write( "variable.  Begin the solution by multiplying the -4 times each of the terms in the parentheses.<BR>\n" );
document.write( "When you do that multiplication the left side of the equation becomes:<BR>\n" );
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document.write( "8x - 8x + 16 = 16<BR>\n" );
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document.write( "Notice that the 8x and the -8x cancel each other out and the equation reduces to:<BR>\n" );
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document.write( "16 = 16<BR>\n" );
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document.write( "Well, that is an interesting development. What does it mean. To find out, let's go back<BR>\n" );
document.write( "to the original pair of equations. Let's take the first equation:<BR>\n" );
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document.write( "8x - 4y = 16<BR>\n" );
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document.write( "and let's try to work it into the form of the second equation ... with just a y on the left<BR>\n" );
document.write( "side and everything else on the right side. Begin by subtracting 8x from both sides to<BR>\n" );
document.write( "get rid of the 8x on the left side. When you do this subtraction the equation becomes:<BR>\n" );
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document.write( "-4y = -8x + 16<BR>\n" );
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document.write( "Now divide both sides of the equation (all terms) by -4 so that you just have y on the left<BR>\n" );
document.write( "side. When you divide all terms on both sides by -4 the equation becomes:<BR>\n" );
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document.write( "y = 2x - 4<BR>\n" );
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document.write( "Wow! Notice that this is the exact same equation as the second equation. Therefore,<BR>\n" );
document.write( "if you graphed both equations you would find that the two graphs coincide ... one graph is<BR>\n" );
document.write( "identical to the other graph.  This means that there is not a unique solution to the two original<BR>\n" );
document.write( "equations. Every solution pair of one equation is also a solution pair of the other equation.<BR>\n" );
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document.write( "Let's try a point just to verify this. For example, suppose in the first equation we <BR>\n" );
document.write( "assume that x = 0. This means that in the equation<BR>\n" );
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document.write( "8x - 4y = 16<BR>\n" );
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document.write( "we set x equal to zero and the equation reduces to<BR>\n" );
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document.write( "-4y = 16<BR>\n" );
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document.write( "Divide both sides by -4 and the equation reduces to<BR>\n" );
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document.write( "y = -4<BR>\n" );
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document.write( "This tells us that the point (0, -4) is on that graph<BR>\n" );
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document.write( "Now go to the second equation and set x = 0. The second equation equation is<BR>\n" );
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document.write( "y = 2x - 4<BR>\n" );
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document.write( "and when you set x = 0 it reduces to <BR>\n" );
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document.write( "y = -4<BR>\n" );
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document.write( "This tells you that the coordinate point (0, -4) also in the solution set for this equation.<BR>\n" );
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document.write( "You can do the same sort of exercise for other values of x and you will always find the<BR>\n" );
document.write( "same thing to be true. For example if you set x equal to 5 in both equations you will find the<BR>\n" );
document.write( "corresponding value of y will be 6. Therefore, the point (5, 6) is a solution for both of<BR>\n" );
document.write( "the equations. There are an infinite number of common solutions, not a just a single solution.<BR>\n" );
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document.write( "In working this problem you came up with the same solution as I did. And you were correct. <BR>\n" );
document.write( "You just needed to find out what this meant when you worked the two equations out and found<BR>\n" );
document.write( "they were identical. Rest easy ... you know what you are doing.<BR>\n" );
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document.write( "Hope this helps you to understand what was going on in this problem.<BR>\n" );
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