document.write( "Question 73044: if the graph of two linear equations in a system have different slopes will the system always have exactly one solution? \n" ); document.write( "
Algebra.Com's Answer #52307 by bucky(2189)\"\" \"About 
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This is true if you don't place any limits on the domain of x or the range of y.
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\n" ); document.write( "There are three possibilities that can occur with linear equations. They are:
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\n" ); document.write( "(1) The first possibility is that the graphs are parallel and different lines. This implies
\n" ); document.write( "that they have the same slopes but are separated by some amount of vertical distance.\r
\n" ); document.write( "\n" ); document.write( "Example:
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\n" ); document.write( "y = 3x + 7
\n" ); document.write( "y = 3x - 2
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\n" ); document.write( "These two equations have parallel graphs because they both have a slope of +3,
\n" ); document.write( "but they
\n" ); document.write( "are separated by 9 vertical units. (One crosses the y-axis at +7 and the other crosses
\n" ); document.write( "the y-axis at -2). In this possibility, because the graphs are parallel, there is no common
\n" ); document.write( "solution.
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\n" ); document.write( "(2) The second possibility is that the graphs lie on top of each other so that every solution
\n" ); document.write( "of one of the equations is a solution of the other equation also.
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\n" ); document.write( "Example:
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\n" ); document.write( "y = 5x = 14
\n" ); document.write( "2y = 10x + 28
\n" ); document.write( "y = (1/2)*(10x + 28)
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\n" ); document.write( "With a little manipulation you will find that these three are all the same equation and
\n" ); document.write( "therefore their graphs are on top of each other. So a solution for one of them is a solution
\n" ); document.write( "for all of them.
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\n" ); document.write( "(3) The third possibility is that the graphs have different slopes. Two such lines can
\n" ); document.write( "intersect at only one point, and that point is the common solution. However, if you put limits
\n" ); document.write( "on the values of x and y in addition to the equation, then there may not be a common solution.
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\n" ); document.write( "Example:
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\n" ); document.write( "Given the equations:
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\n" ); document.write( "y = 2x + 4 and
\n" ); document.write( "y = x + 5
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\n" ); document.write( "If you work this out you will find that the common solution is x = 1 and y = 6.
\n" ); document.write( "(Note: these
\n" ); document.write( "two equations have different slopes. One has a slope of 2 and the other has a slope of +1.)
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\n" ); document.write( "But if the problem had said:
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\n" ); document.write( "Find the common solution of:
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\n" ); document.write( "y = 2x + 4 and
\n" ); document.write( "y = x + 5
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\n" ); document.write( "If the x value must be negative and the y value cannot be less than 7. In that case there
\n" ); document.write( "is no common solution because we know that in the common solution the x value is positive 1
\n" ); document.write( "and the y value is positive 6. Both of these limits (the x value limit and the y value limit)
\n" ); document.write( "cannot be satisfied by the common solution values for x and y.
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\n" ); document.write( "Hope this last part doesn't confuse you. For beginners it is probably enough to say that
\n" ); document.write( "with different slopes there can and will be only one common solution to two linear
\n" ); document.write( "equations.\r
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