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Question 1135207: My math teacher has asked us this one complex question for points, just for fun, but it counts for points. Can someone help? I'm so stressed out!
Consider an equation of a line written in standard form: 4x - 2y = 8. Also consider a point (4, 2) that is not on that line.
(a) Find the equation of the line that is parallel to 4x - 2y = 8 and passes through the point ( 4, 2).
Write the equation is all four forms: slope-intercept, point-slope, standard, and two-point forms.
(b) Find the equation of the line that is perpendicular to 4x - 2y = 8 and passes through the point ( 4, 2).
Write the equation is all four forms: slope-intercept, point-slope, standard, and two-point forms.
(c) Graph all three lines on the same coordinate axis.
For all three lines, graph four (4) points for each line; twelve (12) points and three (3) lines total.
The first point is (4, 2), the second point is of your choosing, and the third & fourth points are
the two intercepts (vertical and horizontal); note that (4, 2) is not on the original given line so pick another point on the first line.
Found 4 solutions by josgarithmetic, rothauserc, MathLover1, ikleyn:
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website! ----------
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My math teacher has asked us this one complex question for points, just for fun, but it counts for points. Can someone help? I'm so stressed out!
Consider an equation of a line written in standard form: 4x - 2y = 8. Also consider a point (4, 2) that is not on that line.
(a) Find the equation of the line that is parallel to 4x - 2y = 8 and passes through the point ( 4, 2).
Write the equation is all four forms: slope-intercept, point-slope, standard, and two-point forms.
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The parallel line will have the same slope as the given line, and this also means that as given in the standard form equation, the left member will be the same. The right-member will be different.
but this can also be simplified, multiplying the two members by  :
 ---------standard form.
Solving for y:
 --------slope-intercept form; and note, the slope is  .
 ----------the point-slope form equation; which comes directly from the formula for slope of a line through two points.
Answer by rothauserc(4718)  (Show Source):
You can put this solution on YOUR website! To begin with, relax :-)
:
slope intercept form is
:
y = mx +b, where m is the slope and b is the y axis intercept
:
point slope form is
:
(y-y1) = m(x-x1), m is the slope and the line passes through the point (x1,y1)
:
standard form is
:
ax +by +c = 0, where a, b, c are constants
:
two point form is
:
(x-x1)/(x2-x1) = (y-y1)/(y2-y1), where the two points are (x1,y1) and (x2,y2)
:
1) 4x - 2y = 8
:
lets write the point slope form of this line
:
-2y = 8 -4x
:
divide both sides of the = by -2
:
2) y = 2x +8
:
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a) the line parallel to equation 2 has the same slope
:
3) y = 2x +b
:
we are given the point (4,2) which the parallel line passes through
:
substitute x = 4 and y = 2 in equation 3
:
2 = 2(4) + b
:
2 = 8 +b
:
b = -6
:
slope intercept form is
:
y = 2x -6
:
point slope form is
:
(y-2) = 2(x-4)
:
standard form is
:
2x -y -6 = 0
:
two point form is
:
use the slope intercept form to find the y axis intercept point
:
set x = 0, then
:
y = 2(0) -6 = -6
:
therefore the second point is (0, -6)
:
two points we are using are (4,2) and (0,-6)
:
(x-4)/(0-4) = (y-2)/(-6-2)
:
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b) the slope of the perpendicular line to equation 2 is the negative
reciprocal of equation 2's slope
:
therefore, the slope is -1/2
:
y = -x/2 +b
:
2 = -4/2 +b
:
b = 4
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slope intercept form is
:
y = -x/2 +4
:
point slope form is
:
(y-2) = -(x-4)/2
:
standard form is
:
x/2 +y -4 = 0
:
the points to use are (4,2) and (0,4)
:
two point form is
:
(x-4)/(0-4) = (y-2)/(4-2)
:
**************************************************************
c) I will use the slope intercept form for the three lines
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4) y = 2x +8
:
points are (1,10),(2,12),(0,8),(-4,0)
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5) y = 2x -6
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points are (4,2),(2,-2),(0,-6),(3,0)
:
6) y = -x/2 +4
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points are (4,2),(2,5),(0,4),(8,0)
:
here is the graph, red line is equation 4, green line is equation 5, and blue line is equation 6
:
:
Answer by MathLover1(20849)  (Show Source):
You can put this solution on YOUR website! (a) Find the equation of the line that is parallel to  and passes through the point ( , ).
recall: parallel lines have same slope
 ...write in slope intercept form; means  where  is aslope and  is y-intercept
 =>slope-intercept form:  and 
now use point-slope form and given point to find equation of the parallel line
 and point ( , ) => and 
 ->point-slope form
 =>slope-intercept form of your equation of the parallel line
 ->standard form
a line passing through the points (, ) and ( ,  ).
 ,  ,
 ,  -> your y-intercept
Linear equation passing through two point is:
 -> two-point form
(b) Find the equation of the line that is perpendicular to and passes through the point ( , ).
recall: perpendicular lines have slopes negative reciprocal to each other
=>slope-intercept form: => perpendicular line will have a slope
now use point-slope form and given point to find equation of the parallel line
 , and point (, ) =>
 ->slope-intercept
 ->standard form
, ,
,  -> your y-intercept
Linear equation passing through two point is:
-> two-point form
(c) Graph all three lines on the same coordinate axis.
Answer by ikleyn(52781)  (Show Source):
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