SOLUTION: 1. Write the equation of the line through (1,2) and (-3,6) in the y=mx+b format. 2.. Solve x + y - z = 0 2x - y + z = 6 -x + y + z = 8 Show your answer

Algebra ->  Expressions-with-variables -> SOLUTION: 1. Write the equation of the line through (1,2) and (-3,6) in the y=mx+b format. 2.. Solve x + y - z = 0 2x - y + z = 6 -x + y + z = 8 Show your answer      Log On


   

Question 83597: 1. Write the equation of the line through (1,2) and (-3,6) in the y=mx+b format.
2.. Solve
x + y - z = 0
2x - y + z = 6
-x + y + z = 8
Show your answers as (x,y,z)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
1.
Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (1,2) and (-3,6)


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 (1,2) and (x%5B2%5D,y%5B2%5D) is the second point (-3,6))


m=%286-2%29%2F%28-3-1%29 Plug in y%5B2%5D=6,y%5B1%5D=2,x%5B2%5D=-3,x%5B1%5D=1 (these are the coordinates of given points)


m=+4%2F-4 Subtract the terms in the numerator 6-2 to get 4. Subtract the terms in the denominator -3-1 to get -4




m=-1 Reduce



So the slope is

m=-1





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



y-2=-x%2B%28-1%29%28-1%29 Distribute -1


y-2=-x%2B1 Multiply -1 and -1 to get 1%2F1. Now reduce 1%2F1 to get 1

y=-x%2B1%2B2 Add 2 to both sides to isolate y


y=-x%2B3 Combine like terms 1 and 2 to get 3

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Answer:



So the equation of the line which goes through the points (1,2) and (-3,6) is:y=-x%2B3


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-1 and the y-intercept is b=3


Notice if we graph the equation y=-x%2B3 and plot the points (1,2) and (-3,6), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=-x%2B3 through the points (1,2) and (-3,6)


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





2.
Start with the given system and label the equations
x+%2B+y+-+z+=+0 Equation 1
2x+-+y+%2B+z+=+6 Equation 2
-x+%2B+y+%2B+z+=+8 Equation 3
3x=6 Add equations 1 and 2
x=2 Divide both sides by 3

Now lets use this info to find another variable

2%2A2-y%2Bz=6 Plug in x=2 into equation 2 to get Equation 4
4-y%2Bz=6 Multiply
-2%2By%2Bz=8 Plug in x=2 into equation 3 to get Equation 5
-%28-2%2By%2Bz%29=-8 Multiply equation 3 by -1

So now we have the equations
4-y%2Bz=6 Equation 4
2-y-z=-8 Equation 5
6-2y=-2 Add equations 4 and 5
-2y=-8 Subtract 6 from both sides
y=4 Divide both sides by -2

Now lets use this to find z

2%2B4-z=0 Plug in x=2 and y=4 into Equation 1 (at this point any equation will do)
6-z=0 Add
-z=-6 Subtract
z=6 Divide

So our solution is
(2,4,6)