SOLUTION: These 3 problems do not come from a book but I really need help with them! If you can solve them as soon as possible, I would greatly appreciate it. Here they are: #1: Write t

Algebra ->  Linear-equations -> SOLUTION: These 3 problems do not come from a book but I really need help with them! If you can solve them as soon as possible, I would greatly appreciate it. Here they are: #1: Write t      Log On


   

Question 98453: These 3 problems do not come from a book but I really need help with them!
If you can solve them as soon as possible, I would greatly appreciate it.
Here they are:
#1:
Write the equation of the line passing through the points listed below. Write the final answer in the slope-intercept form y = mx + b. If the slope (m) or the y-intercept (b) are not integers, fractions must be used.
(-1, 8); (9, 2)

#2:
A line passes through the origin and the point (-6, 4).
Write the equation of the line that passes through (-6, 4) and is perpendicular to the given line.
Write the final answer in the slope-intercept form y = mx + b.
(The slope and y-intercept must be written as fractions, when needed).

#3:
Write the equation for the line perpendicular to
2x - 5y = 5 and passes through the point (-2, -5).
Enter the variable expression Ax + By separately from the constant term C.
Your response should be given in standard form where A, B and C are the smallest possible integers and A > 0.
Thanks so much.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
These explanations are quite lengthy, so here's a mini table of contents

#1
#2
#3



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"#1:
Write the equation of the line passing through the points listed below. Write the final answer in the slope-intercept form y = mx + b. If the slope (m) or the y-intercept (b) are not integers, fractions must be used.
(-1, 8); (9, 2)"



First lets find the slope through the points (-1,8) and (9,2)

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: is the first point (-1,8) and is the second point (9,2))

m=%282-8%29%2F%289--1%29 Plug in y%5B2%5D=2,y%5B1%5D=8,x%5B2%5D=9,x%5B1%5D=-1 (these are the coordinates of given points)

m=+-6%2F10 Subtract the terms in the numerator 2-8 to get -6. Subtract the terms in the denominator 9--1 to get 10


m=-3%2F5 Reduce

So the slope is
m=-3%2F5

------------------------------------------------


Now let's use the point-slope formula to find the equation of the line:



------Point-Slope Formula------
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and is one of the given points

So lets use the Point-Slope Formula to find the equation of the line

y-8=%28-3%2F5%29%28x--1%29 Plug in m=-3%2F5, x%5B1%5D=-1, and y%5B1%5D=8 (these values are given)


y-8=%28-3%2F5%29%28x%2B1%29 Rewrite x--1 as x%2B1


y-8=%28-3%2F5%29x%2B%28-3%2F5%29%281%29 Distribute -3%2F5

y-8=%28-3%2F5%29x-3%2F5 Multiply -3%2F5 and 1 to get -3%2F5

y=%28-3%2F5%29x-3%2F5%2B8 Add 8 to both sides to isolate y

y=%28-3%2F5%29x%2B37%2F5 Combine like terms -3%2F5 and 8 to get 37%2F5 (note: if you need help with combining fractions, check out this solver)


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Answer:


So the equation of the line which goes through the points (-1,8) and (9,2) is:y=%28-3%2F5%29x%2B37%2F5

The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-3%2F5 and the y-intercept is b=37%2F5

Notice if we graph the equation y=%28-3%2F5%29x%2B37%2F5 and plot the points (-1,8) and (9,2), we get this: (note: if you need help with graphing, check out this solver)

Graph of y=%28-3%2F5%29x%2B37%2F5 through the points (-1,8) and (9,2)

Notice how the two points lie on the line. This graphically verifies our answer.



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#2:

"A line passes through the origin and the point (-6, 4).
Write the equation of the line that passes through (-6, 4) and is perpendicular to the given line.
Write the final answer in the slope-intercept form y = mx + b.
(The slope and y-intercept must be written as fractions, when needed). "


First find the equation of the line through the points (0,0) (the origin) and (-6,4)


First lets find the slope through the points (0,0) and (-6,4)

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: is the first point (0,0) and is the second point (-6,4))

m=%284-0%29%2F%28-6-0%29 Plug in y%5B2%5D=4,y%5B1%5D=0,x%5B2%5D=-6,x%5B1%5D=0 (these are the coordinates of given points)

m=+4%2F-6 Subtract the terms in the numerator 4-0 to get 4. Subtract the terms in the denominator -6-0 to get -6


m=-2%2F3 Reduce

So the slope is
m=-2%2F3

------------------------------------------------


Now let's use the point-slope formula to find the equation of the line:



------Point-Slope Formula------
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and is one of the given points

So lets use the Point-Slope Formula to find the equation of the line

y-0=%28-2%2F3%29%28x-0%29 Plug in m=-2%2F3, x%5B1%5D=0, and y%5B1%5D=0 (these values are given)


y-0=%28-2%2F3%29x%2B%28-2%2F3%29%280%29 Distribute -2%2F3

y-0=%28-2%2F3%29x%2B0 Multiply -2%2F3 and 0 to get 0%2F3. Now reduce 0%2F3 to get 0

y=%28-2%2F3%29x%2B0%2B0 Add 0 to both sides to isolate y

y=%28-2%2F3%29x%2B0 Combine like terms 0 and 0 to get 0
------------------------------------------------------------------------------------------------------------
Answer:


So the equation of the line which goes through the points (0,0) and (-6,4) is:y=%28-2%2F3%29x

The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-2%2F3 and the y-intercept is b=0

Notice if we graph the equation y=%28-2%2F3%29x and plot the points (0,0) and (-6,4), we get this: (note: if you need help with graphing, check out this solver)

Graph of y=%28-2%2F3%29x through the points (0,0) and (-6,4)

Notice how the two points lie on the line. This graphically verifies our answer.

Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line


Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of -2%2F3, you can find the perpendicular slope by this formula:

m%5Bp%5D=-1%2Fm where m%5Bp%5D is the perpendicular slope


m%5Bp%5D=-1%2F%28-2%2F3%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%283%2F-2%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=3%2F2 Multiply the fractions.


So the perpendicular slope is 3%2F2



So now we know the slope of the unknown line is 3%2F2 (its the negative reciprocal of -2%2F3 from the line y=%28-2%2F3%29%2Ax%2B0). Also since the unknown line goes through (-6,4), we can find the equation by plugging in this info into the point-slope formula

Point-Slope Formula:

y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope and (x%5B1%5D,y%5B1%5D) is the given point



y-4=%283%2F2%29%2A%28x%2B6%29 Plug in m=3%2F2, x%5B1%5D=-6, and y%5B1%5D=4



y-4=%283%2F2%29%2Ax-%283%2F2%29%28-6%29 Distribute 3%2F2



y-4=%283%2F2%29%2Ax%2B18%2F2 Multiply



y=%283%2F2%29%2Ax%2B18%2F2%2B4Add 4 to both sides to isolate y

y=%283%2F2%29%2Ax%2B18%2F2%2B8%2F2 Make into equivalent fractions with equal denominators



y=%283%2F2%29%2Ax%2B26%2F2 Combine the fractions



y=%283%2F2%29%2Ax%2B13 Reduce any fractions

So the equation of the line that is perpendicular to y=%28-2%2F3%29%2Ax%2B0 and goes through (-6,4) is y=%283%2F2%29%2Ax%2B13


So here are the graphs of the equations y=%28-2%2F3%29%2Ax%2B0 and y=%283%2F2%29%2Ax%2B13




graph of the given equation y=%28-2%2F3%29%2Ax%2B0 (red) and graph of the line y=%283%2F2%29%2Ax%2B13(green) that is perpendicular to the given graph and goes through (-6,4)









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#3


"Write the equation for the line perpendicular to
2x - 5y = 5 and passes through the point (-2, -5).
Enter the variable expression Ax + By separately from the constant term C.
Your response should be given in standard form where A, B and C are the smallest possible integers and A > 0.

First convert 2x - 5y = 5 to slope intercept form


Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)
Convert from standard form (Ax+By = C) to slope-intercept form (y = mx+b)


2x-5y=5 Start with the given equation


2x-5y-2x=5-2x Subtract 2x from both sides


-5y=-2x%2B5 Simplify


%28-5y%29%2F%28-5%29=%28-2x%2B5%29%2F%28-5%29 Divide both sides by -5 to isolate y


y+=+%28-2x%29%2F%28-5%29%2B%285%29%2F%28-5%29 Break up the fraction on the right hand side


y+=+%282%2F5%29x-1 Reduce and simplify


The original equation 2x-5y=5 (standard form) is equivalent to y+=+%282%2F5%29x-1 (slope-intercept form)


The equation y+=+%282%2F5%29x-1 is in the form y=mx%2Bb where m=2%2F5 is the slope and b=-1 is the y intercept.






Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line


Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of 2%2F5, you can find the perpendicular slope by this formula:

m%5Bp%5D=-1%2Fm where m%5Bp%5D is the perpendicular slope


m%5Bp%5D=-1%2F%282%2F5%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%285%2F2%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=-5%2F2 Multiply the fractions.


So the perpendicular slope is -5%2F2



So now we know the slope of the unknown line is -5%2F2 (its the negative reciprocal of 2%2F5 from the line y=%282%2F5%29%2Ax-1). Also since the unknown line goes through (-2,-5), we can find the equation by plugging in this info into the point-slope formula

Point-Slope Formula:

y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope and (x%5B1%5D,y%5B1%5D) is the given point



y%2B5=%28-5%2F2%29%2A%28x%2B2%29 Plug in m=-5%2F2, x%5B1%5D=-2, and y%5B1%5D=-5



y%2B5=%28-5%2F2%29%2Ax%2B%285%2F2%29%28-2%29 Distribute -5%2F2



y%2B5=%28-5%2F2%29%2Ax-10%2F2 Multiply



y=%28-5%2F2%29%2Ax-10%2F2-5Subtract -5 from both sides to isolate y

y=%28-5%2F2%29%2Ax-10%2F2-10%2F2 Make into equivalent fractions with equal denominators



y=%28-5%2F2%29%2Ax-20%2F2 Combine the fractions



y=%28-5%2F2%29%2Ax-10 Reduce any fractions

So the equation of the line that is perpendicular to y=%282%2F5%29%2Ax-1 and goes through (-2,-5) is y=%28-5%2F2%29%2Ax-10


So here are the graphs of the equations y=%282%2F5%29%2Ax-1 and y=%28-5%2F2%29%2Ax-10




graph of the given equation y=%282%2F5%29%2Ax-1 (red) and graph of the line y=%28-5%2F2%29%2Ax-10(green) that is perpendicular to the given graph and goes through (-2,-5)






Now convert %28-5%2F2%29x-10 back to standard form


Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)
Convert from slope-intercept form (y = mx+b) to standard form (Ax+By = C)


y+=+%28-5%2F2%29x-10 Start with the given equation


2%2Ay+=+2%2A%28%28-5%2F2%29x-10%29 Multiply both sides by the LCD 2


2y+=+-5x-20 Distribute and multiply


2y%2B5x+=+-5x-20%2B5x Add 5x to both sides


5x%2B2y+=+-20 Simplify


The original equation y+=+%28-5%2F2%29x-10 (slope-intercept form) is equivalent to 5x%2B2y+=+-20 (standard form where A > 0)


The equation 5x%2B2y+=+-20 is in the form Ax%2BBy+=+C where A+=+5, B+=+2 and C+=+-20








If you have any further questions, feel free to ask.