document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=102508>Question 102508</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the system by addition or substitution.<BR>\n" );
document.write( "	–15x + 3y = 9<BR>\n" );
document.write( "	y = 3 + 5x \r<BR>\n" );
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document.write( "Please help me with this problem step by step if at all possible then maybe I can actually catch on, on how to do these problems.  THANK YOU!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #74555 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=74555\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "–15x + 3y = 9<BR>\n" );
document.write( "y = 3 + 5x <BR>\n" );
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document.write( "You are to solve this by addition or substitution.<BR>\n" );
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document.write( "Let's try by substitution. The object of this method is to take one of the equations and solve<BR>\n" );
document.write( "it for one of the variables in terms of the other variable. Then you substitute that into<BR>\n" );
document.write( "the other equation and solve it. Sort of hard to explain, but easy to do.<BR>\n" );
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document.write( "Look at the bottom equation. It already shows that y equals 3 + 5x. So it is solved for<BR>\n" );
document.write( "y in terms of x. Since we know that y = 3 + 5x, we can go to the top equation and replace y<BR>\n" );
document.write( "by 3 + 5x. When we do that replacement, the top equation becomes:<BR>\n" );
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document.write( "-15x + 3(3 + 5x) = 9<BR>\n" );
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document.write( "On the left side multiply the 3 times both of the terms in the parentheses and the equation<BR>\n" );
document.write( "becomes:<BR>\n" );
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document.write( "-15x + 9 + 15x = 9<BR>\n" );
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document.write( "Wow! Notice that the -15x and the + 15x on the left side cancel each other out and we<BR>\n" );
document.write( "are left with 9 = 9!!! What does that mean? From experience I know that this means that<BR>\n" );
document.write( "there is not just one solution to this problem. This is a trick problem. Let's see why that is true.<BR>\n" );
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document.write( "The top equation is:<BR>\n" );
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document.write( "-15x + 3y = 9<BR>\n" );
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document.write( "and the bottom equation is y = 3 + 5x<BR>\n" );
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document.write( "Let's rearrange the bottom equation so the x and y terms are on the left side and the number<BR>\n" );
document.write( "is on the right side. Do this by getting rid of the 5x on the right side and you can do that<BR>\n" );
document.write( "by subtracting 5x from the right side. But if you subtract 5x from the right side you<BR>\n" );
document.write( "must also subtract 5x from the left side to keep the equation in balance. When you do this<BR>\n" );
document.write( "subtraction, the 5x on the right side disappears and a -5x appears on the left side. Therefore,<BR>\n" );
document.write( "the bottom equation becomes:<BR>\n" );
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document.write( "-5x + y = 3<BR>\n" );
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document.write( "Now we can multiply both sides of this equation by 3 without changing its \"balance\"<BR>\n" );
document.write( "and when we do that the bottom equation then becomes:<BR>\n" );
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document.write( "-15x + 3y = 9<BR>\n" );
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document.write( "Notice anything unusual about that??? It's identical to the original top equation.<BR>\n" );
document.write( "Therefore, it has the same graph as the top equation. This means that every coordinate<BR>\n" );
document.write( "pair on the top equation is also a coordinate pair of the bottom equation. There is not just<BR>\n" );
document.write( "one solution to this problem ... there are many, many solutions that will satisfy both<BR>\n" );
document.write( "equations at the same time.<BR>\n" );
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document.write( "Let's try an example. Suppose in the original bottom equation we let x equal 3. Then <BR>\n" );
document.write( "the bottom equation becomes:<BR>\n" );
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document.write( "y = 3 + 5(3) = 3 + 15 = 18<BR>\n" );
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document.write( "Now let's go to the top equation and let x = 3. When we make that substitution into the<BR>\n" );
document.write( "original top equation, we get:<BR>\n" );
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document.write( "-15(3) + 3y = 9<BR>\n" );
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document.write( "Multiplying out on the left side results in:<BR>\n" );
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document.write( "-45 + 3y = 9<BR>\n" );
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document.write( "Get rid of the -45 by adding 45 to both sides:<BR>\n" );
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document.write( "3y = 54<BR>\n" );
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document.write( "Solve for y by dividing both sides by 3 to get<BR>\n" );
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document.write( "y = 54/3 = 18<BR>\n" );
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document.write( "So the coordinate pair (3, 18) is a common solution for both problems. If you repeat this<BR>\n" );
document.write( "same process for other values of x, you will continue to find that for each x you select and<BR>\n" );
document.write( "substitute into both equations, the corresponding values for y in each of the equations is<BR>\n" );
document.write( "the same.  And whenever you put a common value of x into both equations, the corresponding<BR>\n" );
document.write( "values of y from both equations will be equal. Many solutions ... an infinite number.<BR>\n" );
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document.write( "You would have found out the same thing had you tried to solve the set of equations by <BR>\n" );
document.write( "addition. Not just one unique solution, but many many solutions. <BR>\n" );
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document.write( "Hope this helps you to understand this \"trick\" problem.  Your problem does not have two<BR>\n" );
document.write( "different equations. It just gives you the same equation twice ... in a little different<BR>\n" );
document.write( "form, but nevertheless the same.<BR>\n" );
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