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Question 1016236: Hello,
I am trying to write the slope intercept form of an equation. I figured the equation out, but it's supposed to be perpendicular to an equation already given to me. I took the slope (-1 over 6) from the equation and the two given points it gave me (3, -5). I plugged it in with point slope form and got y= -1/6x - 4.5. But I don't understand how it's perpendicular.
Found 2 solutions by macston, Theo:
Answer by macston(5194)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the line is perpendicular to another line if the slope is a negative reciprocal of that other line.
for example:
if your original equation was y = 6x + 5, then the slope of the line perpendicular to it would be y = -1/6 * x + b, where -1/6 is the negative reciprocal of 6 and b needs to be found using the point given.
the point given appears to be (3,-5).
start with your equation of y = -1/6 * x + b.
you are using the b because you don't know what the y-intercept is yet.
one way to find the value of b is to replace x and y in the equation with 3 and -5.
you will get -5 = -1/6 * 3 + b
simplify to get -5 = -1/2 + b
solve for b to get b = -5 + 1/2 = -4.5
your equation would be y = -1/6 * x - 4.5
assuming your original equation was y = 6x + 5, then your new line will be perpendicular to the original line at the intersection of both lines.
you can graph both lines to see where the intersection is, or you can solve for it algebraically.
i'll graph it for you so you can see what i'm talking about.
your graph is shown below:
the above shows a close up of the intersection and shows the perpendicularity.
the following graph is a more far out show showing the y-intercepts of both equations.
in order to show the perpendicularity, the scale on the graph has to be the same for both the y-axis and the x-axis.
if they are not scaled the same, the lines will not look perpendicular.
these graphs i just showed you above have been squared properly so you can see that the lines are perpendicular to each other.
without the graph,the only way you know the lines are perpendicular to each other is because their slopes are negative reciprocals of each other.
the y-intercept is the point where the lines cross the y-axis.
that's y = 5 for the original equation and y = -4.5 for the perpendicular equation.
the intersect is solved for by solving the equations simultaneously.
your two equations are:
y = 6x + 5
y = -1/6 * x - 4.5
leave the first equation as is and multiply both sides of the second equation by 36 to get:
y = 6x + 5
36y = -6x - 162
add the equations together to get:
37y = -157
divide both sides of this equation by 37 to get:
y = -157/37 = -4.243243243
solve for x in the first equation of y = 6x + 5 to get:
-4.243243243 = 6x + 5
subtract 5 from both sides of this equation to get:
-9.243243243 = 6x
divide both sides of this equation by 6 to get:
-1.540540541 = x
round these to 3 decimal places and you have:
x = -1.541
y = -4.243
the intersection is at the coordinate point of (-1.541,-4.243).
you can see this on the graph.
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