document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=123102>Question 123102</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me?\r<BR>\n" );
document.write( "\n" );
document.write( "Solve 2/5x - 6 = 12\r<BR>\n" );
document.write( "\n" );
document.write( "Solve by addition 5x - 3y = 13<BR>\n" );
document.write( "                  4x - 3y = 11 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #90357 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=90357\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First problem<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Given the following equation to solve:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F5%29x+-+6+=+12 ALIGN=MIDDLE  ALT=\"%282%2F5%29x+-+6+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "You don't have to do this first, but one of the things you can do to simplify this problem is<BR>\n" );
document.write( "to get rid of the denominator 5 by multiplying both sides of the equation (all terms) by 5.<BR>\n" );
document.write( "When you do that multiplication of the first term on the left side, the multiplier 5 cancels with<BR>\n" );
document.write( "the 5 in the denominator and you are left with just 2x for the first term. When you multiply<BR>\n" );
document.write( "the -6 by 5 you get -30, and, on the other side, when you multiply the 12 by 5 you get +60.<BR>\n" );
document.write( "This means that you have converted the given equation to an equivalent form of:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+-+30+=+60 ALIGN=MIDDLE  ALT=\"2x+-+30+=+60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next, the goal is to get the term containing x on one side of the equation and everything else<BR>\n" );
document.write( "on the other side. You can do this by getting rid of the -30 on the left side by adding +30 to<BR>\n" );
document.write( "both sides of the equation. On the left side the addition of +30 cancels the the term -30 out. <BR>\n" );
document.write( "On the right side the addition of +30 to the +60 results in +90 and the equation is then:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+=+90 ALIGN=MIDDLE  ALT=\"2x+=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Solve for x by dividing both sides by the multiplier of the x (that is, divide both sides<BR>\n" );
document.write( "by 2) and you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+90%2F2+=+45 ALIGN=MIDDLE  ALT=\"x+=+90%2F2+=+45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "So the answer is x = 45.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "You can check this answer by returning to the original equation you were given and substituting<BR>\n" );
document.write( "45 for x to see if the equation balances (are both sides equal). The original equations was:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F5%29x+-+6+=+12 ALIGN=MIDDLE  ALT=\"%282%2F5%29x+-+6+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Substitute 45 for x and you have:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F5%29%2A%2845%29+-+6+=+12 ALIGN=MIDDLE  ALT=\"%282%2F5%29%2A%2845%29+-+6+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "One-fifth of 45 is 9 so two-fifths of 45 is 18. Substitute 18 for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F5%29%2A%2845%29 ALIGN=MIDDLE  ALT=\"%282%2F5%29%2A%2845%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18+-+6+=+12 ALIGN=MIDDLE  ALT=\"18+-+6+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "which reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12+=+12 ALIGN=MIDDLE  ALT=\"12+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the equation balances ... so the answer of x = 45 is correct.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Second problem:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Given the two equations:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "+5x - 3y = +13<BR>\n" );
document.write( "+4x - 3y = +11<BR>\n" );
document.write( ".<BR>\n" );
document.write( "To solve these by addition, the object is to get one of the terms in the top equation to<BR>\n" );
document.write( "be equal to but of opposite in sign to the corresponding term in the bottom equation.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Notice that both of the equations have a term -3y in them. But they both have the same sign.<BR>\n" );
document.write( "Suppose that we multiplied the bottom equation by -1 (multiply all terms on both sides). If<BR>\n" );
document.write( "we did that our pair of equations would then be:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "+5x - 3y = +13<BR>\n" );
document.write( "-4x + 3y = -11<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now if we add the two equations vertically (in columns) the sum of the x-terms is 5x - 4x = x.<BR>\n" );
document.write( "The two y-terms are equal but of opposite signs so when you add them they cancel out and disappear.<BR>\n" );
document.write( "And on the right side the +13 and -11 add to give +2. So the sum of the two equations leaves<BR>\n" );
document.write( "you with:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "+x = +2<BR>\n" );
document.write( ".<BR>\n" );
document.write( "To solve for y, just return to either one of the original equations and substitute 2 for x<BR>\n" );
document.write( "and solve for y. Let's return to the original top equation of:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "+5x - 3y = +13<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Substitute 2 for x and this equation becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "5*2 - 3y = +13<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Doing the multiplication 5*2 simplifies this equation to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "10 - 3y = 13<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Get rid of the 10 on the left side by subtracting 10 from both sides to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "-3y = 3<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Solve for y by dividing both sides of the equation by -3 and you have:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "y = 3/(-3) = -1<BR>\n" );
document.write( ".<BR>\n" );
document.write( "So the common solution we have for the set of equations is x = 2, y = -1<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let's check this by going to the bottom equation of the original pair of equations and<BR>\n" );
document.write( "substituting 2 for x and -1 for y to see if the equation balances. The original bottom equation<BR>\n" );
document.write( "was:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "+4x - 3y = +11<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Making the substitutions for x and y results in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "4*2 - 3(-1) = 11<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Multiplying out the two terms on the left side gives:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "8 + 3 = 11<BR>\n" );
document.write( ".<BR>\n" );
document.write( "This reduces to 11 = 11 which indicates that our answers of x = 2 and y = -1 are correct.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this helps to increase your understanding of these two problems.<BR>\n" );
document.write( ".<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );