SOLUTION: find the slope and the y-intercept of the line whose equation is 2x+3y=12 and another question write an equation of the line through (1,-3) and (-2,6)

Algebra ->  Linear-equations -> SOLUTION: find the slope and the y-intercept of the line whose equation is 2x+3y=12 and another question write an equation of the line through (1,-3) and (-2,6)      Log On


   

Question 86175: find the slope and the y-intercept of the line whose equation is 2x+3y=12
and another question
write an equation of the line through (1,-3) and (-2,6)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
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%2B3y=12 Start with the given equation


2x%2B3y-2x=12-2x Subtract 2x from both sides


3y=-2x%2B12 Simplify


%283y%29%2F%283%29=%28-2x%2B12%29%2F%283%29 Divide both sides by 3 to isolate y


y+=+%28-2x%29%2F%283%29%2B%2812%29%2F%283%29 Break up the fraction on the right hand side


y+=+%28-2%2F3%29x%2B4 Reduce and simplify


The original equation 2x%2B3y=12 (standard form) is equivalent to y+=+%28-2%2F3%29x%2B4 (slope-intercept form)


The equation y+=+%28-2%2F3%29x%2B4 is in the form y=mx%2Bb where m=-2%2F3 is the slope and b=4 is the y intercept.





Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (1,-3) and (-2,6)


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


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


m=+9%2F-3 Subtract the terms in the numerator 6--3 to get 9. Subtract the terms in the denominator -2-1 to get -3




m=-3 Reduce



So the slope is

m=-3





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



y%2B3=%28-3%29%28x-1%29 Rewrite y--3 as y%2B3



y%2B3=-3x%2B%28-3%29%28-1%29 Distribute -3


y%2B3=-3x%2B3 Multiply -3 and -1 to get 3%2F1. Now reduce 3%2F1 to get 3

y=-3x%2B3-3 Subtract 3 from both sides to isolate y


y=-3x%2B0 Combine like terms 3 and -3 to get 0

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Answer:



So the equation of the line which goes through the points (1,-3) and (-2,6) is:y=-3x%2B0


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-3 and the y-intercept is b=0


Notice if we graph the equation y=-3x%2B0 and plot the points (1,-3) and (-2,6), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=-3x%2B0 through the points (1,-3) and (-2,6)


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