SOLUTION: write an equation of a line that contains the given point and is perpendicular to the given line 2x + 4y = -7 ; (-3, -2)

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Question 119074: write an equation of a line that contains the given point and is perpendicular to the given line
2x + 4y = -7 ; (-3, -2)

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!



First convert the standard equation 2x%2B4y=-7 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%2B4y=-7 Start with the given equation


2x%2B4y-2x=-7-2x Subtract 2x from both sides


4y=-2x-7 Simplify


%284y%29%2F%284%29=%28-2x-7%29%2F%284%29 Divide both sides by 4 to isolate y


y+=+%28-2x%29%2F%284%29%2B%28-7%29%2F%284%29 Break up the fraction on the right hand side


y+=+%28-1%2F2%29x-7%2F4 Reduce and simplify


The original equation 2x%2B4y=-7 (standard form) is equivalent to y+=+%28-1%2F2%29x-7%2F4 (slope-intercept form)


The equation y+=+%28-1%2F2%29x-7%2F4 is in the form y=mx%2Bb where m=-1%2F2 is the slope and b=-7%2F4 is the y intercept.







Now let's find the equation of the line that is perpendicular to y=%28-1%2F2%29x-7%2F4 which goes through (-3,-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 -1%2F2, 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%28-1%2F2%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%282%2F-1%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=2%2F1 Multiply the fractions.


So the perpendicular slope is 2



So now we know the slope of the unknown line is 2 (its the negative reciprocal of -1%2F2 from the line y=%28-1%2F2%29%2Ax-7%2F4). 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%2B2=2%2A%28x%2B3%29 Plug in m=2, x%5B1%5D=-3, and y%5B1%5D=-2



y%2B2=2%2Ax-%282%29%28-3%29 Distribute 2



y%2B2=2%2Ax%2B6 Multiply



y=2%2Ax%2B6-2Subtract -2 from both sides to isolate y

y=2%2Ax%2B4 Combine like terms

So the equation of the line that is perpendicular to y=%28-1%2F2%29%2Ax-7%2F4 and goes through (-3,-2) is y=2%2Ax%2B4


So here are the graphs of the equations y=%28-1%2F2%29%2Ax-7%2F4 and y=2%2Ax%2B4




graph of the given equation y=%28-1%2F2%29%2Ax-7%2F4 (red) and graph of the line y=2%2Ax%2B4(green) that is perpendicular to the given graph and goes through (-3,-2)