SOLUTION: Consider the line y = 7x-2 . Find the equation of the line that is perpendicular to this line and passes through the point (-3,2) . Find the equation of the line that is para

Algebra ->  Graphs -> SOLUTION: Consider the line y = 7x-2 . Find the equation of the line that is perpendicular to this line and passes through the point (-3,2) . Find the equation of the line that is para      Log On


   

Question 739145: Consider the line y = 7x-2 .
Find the equation of the line that is perpendicular to this line and passes through the point (-3,2) .
Find the equation of the line that is parallel to this line and passes through the point (-3,2) .
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

given:
a line y+=+7x-2 and
a point (-3,2)
1.
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 7, 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%287%2F1%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%281%2F7%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=-1%2F7 Multiply the fractions.


So the perpendicular slope is -1%2F7



So now we know the slope of the unknown line is -1%2F7 (its the negative reciprocal of 7 from the line y=7%2Ax-2). Also since the unknown line goes through (-3,2), 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=%28-1%2F7%29%2A%28x%2B3%29 Plug in m=-1%2F7, x%5B1%5D=-3, and y%5B1%5D=2



y-2=%28-1%2F7%29%2Ax%2B%281%2F7%29%28-3%29 Distribute -1%2F7



y-2=%28-1%2F7%29%2Ax-3%2F7 Multiply



y=%28-1%2F7%29%2Ax-3%2F7%2B2Add 2 to both sides to isolate y

y=%28-1%2F7%29%2Ax-3%2F7%2B14%2F7 Make into equivalent fractions with equal denominators



y=%28-1%2F7%29%2Ax%2B11%2F7 Combine the fractions



y=%28-1%2F7%29%2Ax%2B11%2F7 Reduce any fractions

So the equation of the line that is perpendicular to y=7%2Ax-2 and goes through (-3,2) is y=%28-1%2F7%29%2Ax%2B11%2F7


So here are the graphs of the equations y=7%2Ax-2 and y=%28-1%2F7%29%2Ax%2B11%2F7




graph of the given equation y=7%2Ax-2 (red) and graph of the line y=%28-1%2F7%29%2Ax%2B11%2F7(green) that is perpendicular to the given graph and goes through (-3,2)




2.
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 7 (its from the slope of y=7%2Ax-2 which is also 7). Also since the unknown line goes through (-3,2), 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=7%2A%28x%2B3%29 Plug in m=7, x%5B1%5D=-3, and y%5B1%5D=2



y-2=7%2Ax-%287%29%28-3%29 Distribute 7



y-2=7%2Ax%2B21 Multiply



y=7%2Ax%2B21%2B2Add 2 to both sides to isolate y

y=7%2Ax%2B23 Combine like terms

So the equation of the line that is parallel to y=7%2Ax-2 and goes through (-3,2) is y=7%2Ax%2B23


So here are the graphs of the equations y=7%2Ax-2 and y=7%2Ax%2B23



graph of the given equation y=7%2Ax-2 (red) and graph of the line y=7%2Ax%2B23(green) that is parallel to the given graph and goes through (-3,2)