SOLUTION: Write the equation of the line L satisfying the given geometric conditions. 1. L has y-intercept (0, -3) and is parallel to the line with equation= 2/3x+1 2. L has y-intercept

Algebra ->  Linear-equations -> SOLUTION: Write the equation of the line L satisfying the given geometric conditions. 1. L has y-intercept (0, -3) and is parallel to the line with equation= 2/3x+1 2. L has y-intercept       Log On


   

Question 119896: Write the equation of the line L satisfying the given geometric conditions.
1. L has y-intercept (0, -3) and is parallel to the line with equation= 2/3x+1
2. L has y-intercept (0, 2) and is perpendicular to the line with equation 2x-3y=6.

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 Parallel or Perpendicular to a Given Line


Since any two parallel lines have the same slope we know the slope of the unknown line is 2%2F3 (its from the slope of y=%282%2F3%29%2Ax%2B1 which is also 2%2F3). Also since the unknown line goes through (0,-3), 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%2B3=%282%2F3%29%2A%28x-0%29 Plug in m=2%2F3, x%5B1%5D=0, and y%5B1%5D=-3



y%2B3=%282%2F3%29%2Ax-%282%2F3%29%280%29 Distribute 2%2F3



y%2B3=%282%2F3%29%2Ax-0%2F3 Multiply



y=%282%2F3%29%2Ax-0%2F3-3Subtract -3 from both sides to isolate y

y=%282%2F3%29%2Ax-0%2F3-9%2F3 Make into equivalent fractions with equal denominators



y=%282%2F3%29%2Ax-9%2F3 Combine the fractions



y=%282%2F3%29%2Ax-3 Reduce any fractions

So the equation of the line that is parallel to y=%282%2F3%29%2Ax%2B1 and goes through (0,-3) is y=%282%2F3%29%2Ax-3


So here are the graphs of the equations y=%282%2F3%29%2Ax%2B1 and y=%282%2F3%29%2Ax-3



graph of the given equation y=%282%2F3%29%2Ax%2B1 (red) and graph of the line y=%282%2F3%29%2Ax-3(green) that is parallel to the given graph and goes through (0,-3)











#2






First convert the standard equation 2x-3y=6 into 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-3y=6 Start with the given equation


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


-3y=-2x%2B6 Simplify


%28-3y%29%2F%28-3%29=%28-2x%2B6%29%2F%28-3%29 Divide both sides by -3 to isolate y


y+=+%28-2x%29%2F%28-3%29%2B%286%29%2F%28-3%29 Break up the fraction on the right hand side


y+=+%282%2F3%29x-2 Reduce and simplify


The original equation 2x-3y=6 (standard form) is equivalent to y+=+%282%2F3%29x-2 (slope-intercept form)


The equation y+=+%282%2F3%29x-2 is in the form y=mx%2Bb where m=2%2F3 is the slope and b=-2 is the y intercept.







Now let's find the equation of the line that is perpendicular to y=%282%2F3%29x-2 which goes through (0,2)

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%282%2F3%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%283%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=-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=%282%2F3%29%2Ax-2). Also since the unknown line goes through (0,2), 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-2=%28-3%2F2%29%2A%28x-0%29 Plug in m=-3%2F2, x%5B1%5D=0, and y%5B1%5D=2



y-2=%28-3%2F2%29%2Ax%2B%283%2F2%29%280%29 Distribute -3%2F2



y-2=%28-3%2F2%29%2Ax-0%2F2 Multiply



y=%28-3%2F2%29%2Ax-0%2F2%2B2Add 2 to both sides to isolate y

y=%28-3%2F2%29%2Ax-0%2F2%2B4%2F2 Make into equivalent fractions with equal denominators



y=%28-3%2F2%29%2Ax%2B4%2F2 Combine the fractions



y=%28-3%2F2%29%2Ax%2B2 Reduce any fractions

So the equation of the line that is perpendicular to y=%282%2F3%29%2Ax-2 and goes through (0,2) is y=%28-3%2F2%29%2Ax%2B2


So here are the graphs of the equations y=%282%2F3%29%2Ax-2 and y=%28-3%2F2%29%2Ax%2B2




graph of the given equation y=%282%2F3%29%2Ax-2 (red) and graph of the line y=%28-3%2F2%29%2Ax%2B2(green) that is perpendicular to the given graph and goes through (0,2)