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Question 488297: Hi, I am trying to help my daughter with her Algebra and forgot how to solve for the following (y=mx+b) statement. I hope you can help as I have been out of this since my late twenties!
Write the slope-intercept form of an equation of the line that passes through the given point and is perpendicular to the equation given.
(2,2), y= -1/5 x +5
Found 2 solutions by nerdybill, Theo:
Answer by nerdybill(7384)   (Show Source):
You can put this solution on YOUR website! Write the slope-intercept form of an equation of the line that passes through the given point and is perpendicular to the equation given.
(2,2), y= -1/5 x +5
.
The slope of:
y= -1/5 x +5
is -1/5
.
A line perpendicular must have a slope that is the negative reciprocal:
(-1/5)m = -1
(1/5)m = 1
m = 5 (slope of new line)
use this along with the given point (2,2) and plug into the "point-slope" form:
y - y1 = m(x - x1)
y - 2 = 5(x - 2)
y - 2 = 5x - 10
y = 5x - 8 (this is what they're looking for -- in "slope-intercept" form)
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the slope intercept form of the equation of a straight line is y = mx + b
m is the slope
b is the y intercept
the point you are given is:
(x,y) = (2,2)
the equation you are given is:
y = (-1/5)x + 5
the slope is (-1/5)
the y intercept is 5
the line perpendicular to this line will have a slope that is a negative reciprocal of the slope of this line.
if the slope is a/b, then the negative reciprocal of the slope is -b/a
since our slope is (-1/5), then the negative reciprocal is (5/1) which is equal to 5.
the slope of our line is 5 and it passes through the point (2,2)
the equation of our line will be y = mx + b
we replace the m with 5 and we get:
y = 5x + b
to find b, we replace y with 2 and x with 2 and solve for b.
this is because the point (x,y) = (2,2) which means that x = 2 and y = 2 at that point.
we get:
2 = 5*2 + b which becomes:
2 = 10 + b
subtract 10 from both sides of this equation to get:
-8 = b which becomes:
b = -8
our equation becomes:
y = 5x - 8
a graph of both equations will show that our new equation is perpendicular to the original equation and will pass through the point (2,2).
i have created a horizontal line at y = 2 and a vertical line at x = 2 in order to show you that our new line passes through that point.
for your additional information, the 2 lines will intersect at the point where the two equations are equal to each other.
that point will be solved for below.
the 2 equations are:
y = (-1/5)x + 5
y = 5x - 8
we set both these equations equal to each other to get:
(-1/5)x + 5 = 5x - 8
add 8 to both sides of this equation to get:
(-1/5)x + 13 = 5x
add (1/5)x to both sides of this equation to get:
13 = 5x + (1/5)x which becomes:
13 = 26/5x
that makes x = (5*13)/26 = 5/2)
this is the same as x = 2.5
we'll solve for y in both equations.
the answers should be the same.
y = -(1/5)x + 5 becomes y = -(1/5)*2.5 + 5 which becomes y = 4.5
y = 5x - 8 becomes y = 5*2.5 - 8 which becomes y = 4.5
we have our point of intersection of the 2 lines at x = 2.5 and y = 4.5
this can be expressed as (x,y) = 2.5,4.5)
the graph is shown below with the vertical and horizontal lines removed so you can see the intersection point easier.
i also made the x and y axis symmetrical so you can see that the lines are really perpendicular to each other easier.
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