SOLUTION: Hello out there,I need some help with some slope problems that is giving me a hard time,the first one is,Write the point-slope form of the line that passes through (3,-3)and (5,1).

Algebra ->  Linear-equations -> SOLUTION: Hello out there,I need some help with some slope problems that is giving me a hard time,the first one is,Write the point-slope form of the line that passes through (3,-3)and (5,1).      Log On


   

Question 82171: Hello out there,I need some help with some slope problems that is giving me a hard time,the first one is,Write the point-slope form of the line that passes through (3,-3)and (5,1).The second one is,Write an equation in slope-intercept form of the line having the given slope and passing through the given point:m=o,(0,-3).The next one is,Write the equation in slope-intercept form of the passing through each pair of points:(6,-2)and (3,-4).The last one is,Write the equation in slope-intercept form of the line passing through each pair of points:(-8,2)and (0,6),thanks for whoever can help.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
note: if you need more help with finding an equation through 2 points check out this solver

"Write the point-slope form of the line that passes through (3,-3)and (5,1)"

Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (3,-3) and (5,1)


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (3,-3) and (x%5B2%5D,y%5B2%5D) is the second point (5,1))


m=%281--3%29%2F%285-3%29 Plug in y%5B2%5D=1,y%5B1%5D=-3,x%5B2%5D=5,x%5B1%5D=3 (these are the coordinates of given points)


m=+4%2F2 Subtract the terms in the numerator 1--3 to get 4. Subtract the terms in the denominator 5-3 to get 2




m=2 Reduce



So the slope is

m=2





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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 (x%5B1%5D,y%5B1%5D) is one of the given points


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


y--3=%282%29%28x-3%29 Plug in m=2, x%5B1%5D=3, and y%5B1%5D=-3 (these values are given)



y%2B3=%282%29%28x-3%29 Rewrite y--3 as y%2B3



y%2B3=2x%2B%282%29%28-3%29 Distribute 2


y%2B3=2x-6 Multiply 2 and -3 to get -6%2F1. Now reduce -6%2F1 to get -6

y=2x-6-3 Subtract 3 from both sides to isolate y


y=2x-9 Combine like terms -6 and -3 to get -9

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Answer:



So the equation of the line which goes through the points (3,-3) and (5,1) is:y=2x-9


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=2 and the y-intercept is b=-9


Notice if we graph the equation y=2x-9 and plot the points (3,-3) and (5,1), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=2x-9 through the points (3,-3) and (5,1)


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






"The second one is,Write an equation in slope-intercept form of the line having the given slope and passing through the given point:m=0,(0,-3)"


Solved by pluggable solver: FIND a line by slope and one point

What we know about the line whose equation we are trying to find out:

  • it goes through point (0, -3)

  • it has a slope of 0



First, let's draw a diagram of the coordinate system with point (0, -3) plotted with a little blue dot:



Write this down: the formula for the equation, given point x%5B1%5D%2C+y%5B1%5D and intercept a, is

y=ax+%2B+%28y%5B1%5D-a%2Ax%5B1%5D%29 (see a paragraph below explaining why this formula is correct)

Given that a=0, and system%28+x%5B1%5D+=+0%2C+y%5B1%5D+=+-3+%29+, we have the equation of the line:

y=0%2Ax+%2B+-3

Explanation: Why did we use formula y=ax+%2B+%28y%5B1%5D+-+a%2Ax%5B1%5D%29 ? Explanation goes here. We are trying to find equation y=ax+b. The value of slope (a) is already given to us. We need to find b. If a point (x%5B1%5D, y%5B1%5D) lies on the line, it means that it satisfies the equation of the line. So, our equation holds for (x%5B1%5D, y%5B1%5D): y%5B1%5D+=+a%2Ax%5B1%5D%2Bb Here, we know a, x%5B1%5D, and y%5B1%5D, and do not know b. It is easy to find out: b=y%5B1%5D-a%2Ax%5B1%5D. So, then, the equation of the line is: +y=ax%2B%28y%5B1%5D-a%2Ax%5B1%5D%29+.

Here's the graph:





note: its hard to see, but its a horizontal line going through (0,-3)

"The next one is,Write the equation in slope-intercept form of the passing through each pair of points:(6,-2)and (3,-4)"

Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (6,-2) and (3,-4)


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (6,-2) and (x%5B2%5D,y%5B2%5D) is the second point (3,-4))


m=%28-4--2%29%2F%283-6%29 Plug in y%5B2%5D=-4,y%5B1%5D=-2,x%5B2%5D=3,x%5B1%5D=6 (these are the coordinates of given points)


m=+-2%2F-3 Subtract the terms in the numerator -4--2 to get -2. Subtract the terms in the denominator 3-6 to get -3




m=2%2F3 Reduce



So the slope is

m=2%2F3





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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 (x%5B1%5D,y%5B1%5D) is one of the given points


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


y--2=%282%2F3%29%28x-6%29 Plug in m=2%2F3, x%5B1%5D=6, and y%5B1%5D=-2 (these values are given)



y%2B2=%282%2F3%29%28x-6%29 Rewrite y--2 as y%2B2



y%2B2=%282%2F3%29x%2B%282%2F3%29%28-6%29 Distribute 2%2F3


y%2B2=%282%2F3%29x-4 Multiply 2%2F3 and -6 to get -12%2F3. Now reduce -12%2F3 to get -4

y=%282%2F3%29x-4-2 Subtract 2 from both sides to isolate y


y=%282%2F3%29x-6 Combine like terms -4 and -2 to get -6

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Answer:



So the equation of the line which goes through the points (6,-2) and (3,-4) is:y=%282%2F3%29x-6


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=-6


Notice if we graph the equation y=%282%2F3%29x-6 and plot the points (6,-2) and (3,-4), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=%282%2F3%29x-6 through the points (6,-2) and (3,-4)


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





"Write the equation in slope-intercept form of the line passing through each pair of points:(-8,2)and (0,6)"

Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (-8,2) and (0,6)


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (-8,2) and (x%5B2%5D,y%5B2%5D) is the second point (0,6))


m=%286-2%29%2F%280--8%29 Plug in y%5B2%5D=6,y%5B1%5D=2,x%5B2%5D=0,x%5B1%5D=-8 (these are the coordinates of given points)


m=+4%2F8 Subtract the terms in the numerator 6-2 to get 4. Subtract the terms in the denominator 0--8 to get 8




m=1%2F2 Reduce



So the slope is

m=1%2F2





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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 (x%5B1%5D,y%5B1%5D) is one of the given points


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


y-2=%281%2F2%29%28x--8%29 Plug in m=1%2F2, x%5B1%5D=-8, and y%5B1%5D=2 (these values are given)



y-2=%281%2F2%29%28x%2B8%29 Rewrite x--8 as x%2B8



y-2=%281%2F2%29x%2B%281%2F2%29%288%29 Distribute 1%2F2


y-2=%281%2F2%29x%2B4 Multiply 1%2F2 and 8 to get 8%2F2. Now reduce 8%2F2 to get 4

y=%281%2F2%29x%2B4%2B2 Add 2 to both sides to isolate y


y=%281%2F2%29x%2B6 Combine like terms 4 and 2 to get 6

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Answer:



So the equation of the line which goes through the points (-8,2) and (0,6) is:y=%281%2F2%29x%2B6


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=1%2F2 and the y-intercept is b=6


Notice if we graph the equation y=%281%2F2%29x%2B6 and plot the points (-8,2) and (0,6), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=%281%2F2%29x%2B6 through the points (-8,2) and (0,6)


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