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You have two unknowns and therefore you need at least two independent equations to be able to
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\n" ); document.write( "One of the ways to solve for two equations with two unknowns is to make one of the equations have
\n" ); document.write( "a term in it that is equal in size but has the opposite sign as the corresponding term in the
\n" ); document.write( "other equation. Then add the two equations together in vertical columns and one of the unknown
\n" ); document.write( "terms drops out. That leaves you with a new equation that has only one variable, and you can
\n" ); document.write( "solve for that variable. Once you get one of the variables, you can go back to one of the original
\n" ); document.write( "equations, substitute for that variable, and solve for the second variable. Sounds complicated,
\n" ); document.write( "but it's not too bad ... let's just do the problem step by step.
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\n" ); document.write( "Given the two equations:
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\n" ); document.write( "4x + 5y = -14
\n" ); document.write( "8x + 10y = -20
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\n" ); document.write( "Let's try to get rid of the variable x. We can do that by multiplying the top equation
\n" ); document.write( "(both sides ... all terms) by -2. This changes the top equation to:
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\n" ); document.write( "-8x - 10y = +28
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\n" ); document.write( "and when we put this over the original bottom equation the pair of equations is then:
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\n" ); document.write( "-8x - 10y = +28
\n" ); document.write( "8x + 10y = -20
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\n" ); document.write( "Notice now what happens when we add these two equations vertically in columns. The -8x
\n" ); document.write( "in the top equation cancels the +8x in the bottom equation ... so there is no x term in
\n" ); document.write( "the resulting equation.
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\n" ); document.write( "But this problem also has a trick in it. Notice that when you add the terms in the y column,
\n" ); document.write( "the -10y and the +10y also cancels out. So we can't solve this problem as we normally would.
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\n" ); document.write( "Since both the x-terms and the y-terms cancel out and the two terms on the right side are different ...
\n" ); document.write( "these two equations are for parallel lines. On a graph they will never cross and therefore,
\n" ); document.write( "there is no point of the form (x,y) which will satisfy both equations.
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\n" ); document.write( "Not a good problem to start with if you are having trouble understanding the concept of
\n" ); document.write( "solving two equations with two unknowns. Just remember the process above and post a new
\n" ); document.write( "example that might have a common solution.
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\n" ); document.write( "Sorry for the confusion above, but you've seen an example of a set of two linear equations in which
\n" ); document.write( "both variables disappear if you try variable elimination. And if that ever happens to you
\n" ); document.write( "again you now know what it means.
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