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Given:
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\n" ); document.write( "+x + 3y = 11
\n" ); document.write( "6x - 2y = 7
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\n" ); document.write( "There are multiple ways that this can be solved (variable elimination, substitution, graphing,
\n" ); document.write( "and Cramer's rule are a few of the ways). Let's use variable elimination. The goal of using
\n" ); document.write( "this method is to get rid of one of the vertical columns by making one of the terms in
\n" ); document.write( "the top equation equal but opposite in sign to the term in the bottom equation that is
\n" ); document.write( "in the same column. Then add the columns vertically to get another equation, and solve it.
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\n" ); document.write( "For example, in this problem we could multiply the top equation (all terms on both sides) by
\n" ); document.write( "+2 and when we do that the equation set becomes:
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\n" ); document.write( "+2x + 6y = +22
\n" ); document.write( "+6x - 2y = + 7
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\n" ); document.write( "Now multiply the bottom equation (all terms on both sides) by +3 and the equation set
\n" ); document.write( "then becomes:
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\n" ); document.write( "+ 2x + 6y = +22
\n" ); document.write( "+18x - 6y = +21
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\n" ); document.write( "Now add the two equations vertically. Note that the +6y and the -6y cancel, and this eliminates
\n" ); document.write( "the y-column. The resulting equation after the vertical addition is:
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\n" ); document.write( "+20x = 43
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\n" ); document.write( "Solve for x by dividing both sides by 20 to get:
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\n" ); document.write( "x = 43/20
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\n" ); document.write( "Next you can solve for y by returning to one of the two original equations and substituting
\n" ); document.write( "43/20 for x. Lets return to the equation:
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\n" ); document.write( "x + 3y = 11
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\n" ); document.write( "substitute 43/20 for x and this equation becomes:
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\n" ); document.write( "43/20 + 3y = 11
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\n" ); document.write( "Get rid of the denominator 20 by multiplying all terms on both sides by 20 to make the
\n" ); document.write( "equation become:
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\n" ); document.write( "43 + 60y = 220
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\n" ); document.write( "Get rid of the 43 on the left side by subtracting 43 from both sides to reduce the equation to:
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\n" ); document.write( "60y = 220 - 43 = 177
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\n" ); document.write( "Solve for y by dividing both sides of this equation by 60:
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\n" ); document.write( "y = 177/60 = (3*59)/(3*20)
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\n" ); document.write( "The 3 in the numerator cancels with the 3 in the denominator to reduce y to:
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\n" ); document.write( "y = 59/20
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\n" ); document.write( "So the answer to this problem is x = 43/20 and y = 59/20
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\n" ); document.write( "Hope this helps you to understand the process of variable elimination by changing both
\n" ); document.write( "of the original equations to eliminate one of the variables.
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