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x + y = 3
\n" ); document.write( "x + y = –1\r
\n" ); document.write( "\n" ); document.write( "This is a \"trick\" question. Let's convert the two equations to the slope-intercept form:
\n" ); document.write( "y=mx+b
\n" ); document.write( "In this form the multiplier of x (which is m) is the slope of the graph and b is the value
\n" ); document.write( "of y at which the graph crosses the y-axis.
\n" ); document.write( "We can convert the top equation to this form by subtracting x from both sides. The
\n" ); document.write( "resulting equation is:
\n" ); document.write( "y = -x + 3
\n" ); document.write( "Compare this equation with the slope-intercept form. Note that the multiplier of x is -1,
\n" ); document.write( "so the slope of the graph is -1. The point at which the graph crosses the y-axis
\n" ); document.write( "is b which in this case is plus 3.
\n" ); document.write( "Now let's re-arrange the second equation into the same slope-intercept form. We do that
\n" ); document.write( "by subtracting x from both sides to get:
\n" ); document.write( "y = -x - 1
\n" ); document.write( "Note that by comparing this equation with the slope-intercept form we again find that
\n" ); document.write( "the slope (the multiplier of x) is -1, but this time the point at which the graph
\n" ); document.write( "crosses the y-axis (that is the point b) is -1.\r
\n" ); document.write( "\n" ); document.write( "What does that tell us? Because the two graphs of these equations have the same slope
\n" ); document.write( "they are parallel!!! The only difference is that one line is higher up (crossing the
\n" ); document.write( "y-axis at y=3) than the other line (crossing the y-axis at y=-1).
\n" ); document.write( "If the pair of equations has a common solution, the two graphs must intersect each other
\n" ); document.write( "at that common point. Since the two graphs are parallel in this case, there never
\n" ); document.write( "intersect. Therefore, there is no point common to the two graphs. There are no
\n" ); document.write( "values
\n" ); document.write( "for x and y that will satisfy both equations. Somebody tried to trick you ... or at least
\n" ); document.write( "wanted you to think about the situation.\r
\n" ); document.write( "\n" ); document.write( "And if you think about it, by looking at the original equations you might have questioned
\n" ); document.write( "how in one case x added to y could give you 3 as an answer and in the very next equation
\n" ); document.write( "the same value for x added to the same value for y could give you -1 as an answer.
\n" ); document.write( "It would have been a clue that something wasn't right.
\n" ); document.write( "Hope this helps your understanding of pairs of linear equations.
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