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Question 71033: Are the following lines parallel, perpendicular, or neither?
L1 with equation x – 2y = 10
L2 with equation 2x + y = 2
Found 2 solutions by checkley75, bucky:
Answer by checkley75(3666)   (Show Source):
You can put this solution on YOUR website! X-2Y=10
-2Y=-X+10
Y=-X/-2+10/-2
Y=X/2-5
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2X+Y=2
Y=-2X+2
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 (graph 300x200 pixels, x from -6 to 5, y from -10 to 10, of TWO functions y = x/2 -5 and y = -2x+2).
checking the two lines ploted I'd say they are neither parallel nor perpendicular.
Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! An easy way to do this problem, given that we are asked if the lines are perpendicular,
or parallel, is to look at the slope of the two given lines. If the slopes are equal, and
the lines do not lie on top of each other, the lines are parallel. If the lines have slopes
such that the slope of one of the lines is the negative inverse of the slope of other line,
the lines are perpendicular.
.
Let's get the lines into the slope intercept form y = mx + b. In this form m is the slope
of the line, and b is the point at which the line crosses the y-axis.
.
Line 1 is given by the equation:
.
x - 2y = 10
.
The slope intercept form requires that the y term be by itself on the left side and the x
term be on the right side of the equation. We can head in this direction by subtracting
x from the left side of the equation to make it disappear. But whatever we do to one side
of the equation, we must also do to the other side. So we must subtract x from the right
side also. When we do these two subtractions (left and right side) the equation becomes:
.
-2y = -x + 10.
.
One more thing to do. Notice the slope intercept form has only y on the left side and
the form we have has -2y on the left side. We can change the left side to y by dividing
the left side by -2. But if we do, then we also have to divide the right side by -2.
We do need just a y on the left side, so we'll divide both sides by -2. When we do the
divisions the equation becomes:
.
.
By comparing this with the slope intercept form we can tell that this line has a slope of
(+1/2) because +1/2 is the multiplier of the x term and since the value of b is -5, the
graphed line crosses the y-axis at -5.
.
Now let's look at the second line. This line has the equation:
.
2x + y = 2
.
To put this into the slope intercept form we subtract 2x from both sides. The resulting
equation is:
.
y = -2x + 2
.
By comparison with the slope intercept form we can tell that this equation is actually in
the slope intercept form and because of that we can also tell that the slope (the multiplier
of x) is -2 and the place where this line crosses the y-axis is at the value +2.
.
So the slope of line 1 is +1/2 and the slope of line 2 is -2. Obviously these two slopes
are not equal. Therefore, the lines are NOT parallel. The next thing to check is are
the lines perpendicular. We need to check to see if we find the negative inverse of the
slope of one of the lines, does that result equal the slope of the other line? Let's find
the negative inverse of line 2. The slope of line 2 is -2. We first take the negative of
that and get +2. Next we find the inverse of that by using it as the denominator in a fraction
that always has the number +1 as the numerator. This makes the negative inverse be (+1/+2)
and that is +(1/2). Notice that the negative inverse of the slope for line 2 is exactly equal
to the slope of line 1. Because of this we can say that line 1 and line 2 are perpendicular
to each other.
.
Work through this example step by step to make sure you understand the process and how
to apply it. Hopefully this will help you to understand how to work similar problems,
and the process will become familiar to you. Another thing you could do is to graph the
two equations and you will see how the lines are perpendicular. Graph each equation
by selecting values for x, plug these values into each equation to get the corresponding
values of y and plot each x-y pair. [I suggest using x=0, x=2, and x=4 in both equations.]
If you do plot the graphs you will see that line 1 slants up as you look to the right and
line 2 slants down ... and the two graphs will look perpendicular to you.
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