document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=132034>Question 132034</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0395937760/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Choosing A Method:<BR>\n" );
document.write( "Solve the linear system.\r<BR>\n" );
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document.write( "8x + 9y = 42<BR>\n" );
document.write( "6x - y = 16\r<BR>\n" );
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document.write( "I can get my first equation to equal 42 with the coordinates (2/5/23,2/16/23) and (1/20/31,3/19/93) <BR>\n" );
document.write( "But I can't get either of those coordinates to fit in the bottom equation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #96431 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=96431\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given the set of equations:<BR>\n" );
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document.write( "8x + 9y = 42<BR>\n" );
document.write( "6x - y = 16<BR>\n" );
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document.write( "There are three ways you can solve these two equations to get a common solution. <BR>\n" );
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document.write( "One way is to graph both equations and find the point where the two linear graphs cross.<BR>\n" );
document.write( "That (x, y) intersection point will give you the values that are common to both equations.<BR>\n" );
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document.write( "Another way to do the problem is to solve one of the equations for one of the variables<BR>\n" );
document.write( "and substitute the equivalent value into the other equation, and solve it.<BR>\n" );
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document.write( "Let's do that method. Note that in the bottom equation, you have a \"-y\" term. If you subtract<BR>\n" );
document.write( "6x from both sides of this bottom equation you have:<BR>\n" );
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document.write( "-y = -6x + 16<BR>\n" );
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document.write( "Then multiply both sides (all terms) by -1 and the equation becomes:<BR>\n" );
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document.write( "y = 6x - 16<BR>\n" );
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document.write( "Next take the right side of this equation (6x - 16) and substitute it for y in the original top<BR>\n" );
document.write( "equation. When you make that substitution in the original top equation you get:<BR>\n" );
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document.write( "8x + 9(6x - 16) = 42<BR>\n" );
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document.write( "Multiply the 9 times each of the terms inside the parentheses and you get:<BR>\n" );
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document.write( "8x + 54x - 144 = 42<BR>\n" );
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document.write( "Combine the two terms containing x ... (8x + 54x = 62x) ... and the equation becomes:<BR>\n" );
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document.write( "62x - 144 = 42<BR>\n" );
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document.write( "Get rid of the -144 on the left side by adding 144 to both sides to get:<BR>\n" );
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document.write( "62x = 186<BR>\n" );
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document.write( "Solve for x by dividing both sides of this equation by 62 and the result is:<BR>\n" );
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document.write( "x = 186/62 = 3<BR>\n" );
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document.write( "Now that you know x = 3, you can solve for y by returning to either of the original equations<BR>\n" );
document.write( "and substituting 3 for x. Then solve for y. Let's return to the bottom of the two original equations:<BR>\n" );
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document.write( "6x - y = 16<BR>\n" );
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document.write( "Substitute 3 for x and the equation becomes:<BR>\n" );
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document.write( "6*3 - y = 16<BR>\n" );
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document.write( "Multiplying out 6*3 and you get:<BR>\n" );
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document.write( "18 - y = 16<BR>\n" );
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document.write( "Subtract 18 from both sides:<BR>\n" );
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document.write( "-y = -2<BR>\n" );
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document.write( "Multiply both sides by -1:<BR>\n" );
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document.write( "y = 2<BR>\n" );
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document.write( "So the common solution to both equations is x = 3 and y = 2. This tells you that the (x, y) <BR>\n" );
document.write( "point where the two graphs cross is (3, 2). It also tells you that if you substitute 3<BR>\n" );
document.write( "for x and 2 for y in the two original equations, the left side of each equation should equal the<BR>\n" );
document.write( "right side.<BR>\n" );
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document.write( "A third way to work this problem is by the method of variable elimination. The way to do this<BR>\n" );
document.write( "is to get a term in the top equation and the corresponding term in the bottom equation to be<BR>\n" );
document.write( "equal in value. Then you vertically add or subtract that term to get rid of it. Let's do it.<BR>\n" );
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document.write( "Start with the two original equations:<BR>\n" );
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document.write( "8x + 9y = 42<BR>\n" );
document.write( "6x - y = 16<BR>\n" );
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document.write( "Multiply the bottom equation (both sides all terms) by 9 to make the equation set become:<BR>\n" );
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document.write( "8x + 9y = 42<BR>\n" );
document.write( "54x -9y = 144<BR>\n" );
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document.write( "Now if you add the two equations in vertical columns, note that the +9y and the -9y cancel<BR>\n" );
document.write( "each other out and the sum of the two equations vertically becomes:<BR>\n" );
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document.write( "8x + 9y = 42<BR>\n" );
document.write( "54x -9y = 144<BR>\n" );
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document.write( "62x + 0 = 186<BR>\n" );
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document.write( "Which simplifies to:<BR>\n" );
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document.write( "62x = 186<BR>\n" );
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document.write( "Divide both sides by 62, just as we did in the substitution method and you again get the<BR>\n" );
document.write( "answer x = 3. Solve for y by going back to the original equations and in one of them replace<BR>\n" );
document.write( "x by 3 and solve for y.<BR>\n" );
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document.write( "Hope this helps you to see the ways you can do this problem.<BR>\n" );
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