document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=109368>Question 109368</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0028242270/algebrahomeworkh>algebra2 with trigonometry</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Solve this system using Cramer's rule x-3y=8 and 2x+2y=8. Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #79818 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=79818\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For 2 by 2 determinants, the determinant:<BR>\n" );
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document.write( "| A B |<BR>\n" );
document.write( "| C D |<BR>\n" );
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document.write( "Has a value of (A*D) - (B*C)<BR>\n" );
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document.write( "Set up your two equations as:<BR>\n" );
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document.write( "1x - 3y = 8<BR>\n" );
document.write( "2x + 2y = 8<BR>\n" );
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document.write( "The first determinant is formed by the coefficients of the x and y terms in the arrangement<BR>\n" );
document.write( "that they appear in the two equations:<BR>\n" );
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document.write( "| 1 -3 |<BR>\n" );
document.write( "| 2 +2 |<BR>\n" );
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document.write( "Evaluate this determinant using the pattern shown above and you get:<BR>\n" );
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document.write( "(1*2)-(-3*2) = 2 - (-6) = 2 + 6 = +8<BR>\n" );
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document.write( "Next you can solve for x by replacing the x column in this determinant with the corresponding<BR>\n" );
document.write( "constants on the right side, find the value of this determinant, and divide that value by<BR>\n" );
document.write( "+8 which is the value of the determinant we worked above using the coefficients of the <BR>\n" );
document.write( "variables.<BR>\n" );
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document.write( "First, the new determinant (for x) is:<BR>\n" );
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document.write( "| 8 -3 |<BR>\n" );
document.write( "| 8 +2 |<BR>\n" );
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document.write( "Note that the x column has been replaced by the column of numbers on the right side of<BR>\n" );
document.write( "the two equations. Evaluate this determinant by using the pattern above and you get:<BR>\n" );
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document.write( "(8*2)-(-3*8) = 16 - (-24) = 16 + 24 = 40<BR>\n" );
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document.write( "Solve for x by dividing this result by the +8 we got for the first determinant and you have:<BR>\n" );
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document.write( "x = 40/8 = 5<BR>\n" );
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document.write( "Next you solve for y. Return to our first determinant and replace the y column with the<BR>\n" );
document.write( "column of constants from the right side of the two equations. So starting with:<BR>\n" );
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document.write( "| 1 -3 |<BR>\n" );
document.write( "| 2 +2 |<BR>\n" );
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document.write( "replace the y column of coefficients by the 8 and 8 constants on the right side and you <BR>\n" );
document.write( "have:<BR>\n" );
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document.write( "| 1 +8 |<BR>\n" );
document.write( "| 2 +8 |<BR>\n" );
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document.write( "Using the pattern to solve this determinant you get:<BR>\n" );
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document.write( "(1*8) - (8*2) = 8 - 16 = -8<BR>\n" );
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document.write( "Divide this by the value of the determinant that has only the coefficients of x and y and you get:<BR>\n" );
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document.write( "y = -8/8 = -1<BR>\n" );
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document.write( "So the two answers are x = 5 and y = -1<BR>\n" );
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document.write( "For 2 by 2 determinants you will need to remember the pattern to evaluate them ... multiply<BR>\n" );
document.write( "along the diagonal down and to the right and then subtract from that the product obtained by<BR>\n" );
document.write( "multiplying along the diagonal down and to the left.<BR>\n" );
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document.write( "Then you evaluate 3 determinants ... first the determinant using only the coefficients.<BR>\n" );
document.write( "Next the determinant formed when you replace the column of x coefficients by the column of <BR>\n" );
document.write( "constants. Then the determinant formed when you when you replace the column of y coefficients by the column of <BR>\n" );
document.write( "constants. <BR>\n" );
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document.write( "Finally, you solve for x by dividing its determinant by the first determinant (coefficients only) <BR>\n" );
document.write( "and solve for y by dividing its determinant by the coefficients only determinant.<BR>\n" );
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document.write( "And you can always check your answers by plugging the values for x and y back into the <BR>\n" );
document.write( "original equations, just like you always do, and see if the equations are true ... the<BR>\n" );
document.write( "left sides equal the right sides.<BR>\n" );
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