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An easy way to do this problem, given that we are asked if the lines are perpendicular,
\n" ); document.write( "or parallel, is to look at the slope of the two given lines. If the slopes are equal, and
\n" ); document.write( "the lines do not lie on top of each other, the lines are parallel. If the lines have slopes
\n" ); document.write( "such that the slope of one of the lines is the negative inverse of the slope of other line,
\n" ); document.write( "the lines are perpendicular.
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\n" ); document.write( "Let's get the lines into the slope intercept form y = mx + b. In this form m is the slope
\n" ); document.write( "of the line, and b is the point at which the line crosses the y-axis.
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\n" ); document.write( "Line 1 is given by the equation:
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\n" ); document.write( "x - 2y = 10
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\n" ); document.write( "The slope intercept form requires that the y term be by itself on the left side and the x
\n" ); document.write( "term be on the right side of the equation. We can head in this direction by subtracting
\n" ); document.write( "x from the left side of the equation to make it disappear. But whatever we do to one side
\n" ); document.write( "of the equation, we must also do to the other side. So we must subtract x from the right
\n" ); document.write( "side also. When we do these two subtractions (left and right side) the equation becomes:
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\n" ); document.write( "-2y = -x + 10.
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\n" ); document.write( "One more thing to do. Notice the slope intercept form has only y on the left side and
\n" ); document.write( "the form we have has -2y on the left side. We can change the left side to y by dividing
\n" ); document.write( "the left side by -2. But if we do, then we also have to divide the right side by -2.
\n" ); document.write( "We do need just a y on the left side, so we'll divide both sides by -2. When we do the
\n" ); document.write( "divisions the equation becomes:
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\n" ); document.write( "By comparing this with the slope intercept form we can tell that this line has a slope of
\n" ); document.write( "(+1/2) because +1/2 is the multiplier of the x term and since the value of b is -5, the
\n" ); document.write( "graphed line crosses the y-axis at -5.
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\n" ); document.write( "Now let's look at the second line. This line has the equation:
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\n" ); document.write( "2x + y = 2
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\n" ); document.write( "To put this into the slope intercept form we subtract 2x from both sides. The resulting
\n" ); document.write( "equation is:
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\n" ); document.write( "y = -2x + 2
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\n" ); document.write( "By comparison with the slope intercept form we can tell that this equation is actually in
\n" ); document.write( "the slope intercept form and because of that we can also tell that the slope (the multiplier
\n" ); document.write( "of x) is -2 and the place where this line crosses the y-axis is at the value +2.
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\n" ); document.write( "So the slope of line 1 is +1/2 and the slope of line 2 is -2. Obviously these two slopes
\n" ); document.write( "are not equal. Therefore, the lines are NOT parallel. The next thing to check is are
\n" ); document.write( "the lines perpendicular. We need to check to see if we find the negative inverse of the
\n" ); document.write( "slope of one of the lines, does that result equal the slope of the other line? Let's find
\n" ); document.write( "the negative inverse of line 2. The slope of line 2 is -2. We first take the negative of
\n" ); document.write( "that and get +2. Next we find the inverse of that by using it as the denominator in a fraction
\n" ); document.write( "that always has the number +1 as the numerator. This makes the negative inverse be (+1/+2)
\n" ); document.write( "and that is +(1/2). Notice that the negative inverse of the slope for line 2 is exactly equal
\n" ); document.write( "to the slope of line 1. Because of this we can say that line 1 and line 2 are perpendicular
\n" ); document.write( "to each other.
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\n" ); document.write( "Work through this example step by step to make sure you understand the process and how
\n" ); document.write( "to apply it. Hopefully this will help you to understand how to work similar problems,
\n" ); document.write( "and the process will become familiar to you. Another thing you could do is to graph the
\n" ); document.write( "two equations and you will see how the lines are perpendicular. Graph each equation
\n" ); document.write( "by selecting values for x, plug these values into each equation to get the corresponding
\n" ); document.write( "values of y and plot each x-y pair. [I suggest using x=0, x=2, and x=4 in both equations.]
\n" ); document.write( "If you do plot the graphs you will see that line 1 slants up as you look to the right and
\n" ); document.write( "line 2 slants down ... and the two graphs will look perpendicular to you. \n" ); document.write( "