SOLUTION: How would you write an equation where its point is (8,-3) and its perpendicular to 4x-3y=10? How would you find an equation that matches that on a graph?

Algebra ->  Linear-equations -> SOLUTION: How would you write an equation where its point is (8,-3) and its perpendicular to 4x-3y=10? How would you find an equation that matches that on a graph?      Log On


   

Question 98495: How would you write an equation where its point is (8,-3) and its perpendicular to 4x-3y=10?
How would you find an equation that matches that on a graph?
Found 2 solutions by jim_thompson5910, edjones:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First convert 4x-3y=10 to 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)


4x-3y=10 Start with the given equation


4x-3y-4x=10-4x Subtract 4x from both sides


-3y=-4x%2B10 Simplify


%28-3y%29%2F%28-3%29=%28-4x%2B10%29%2F%28-3%29 Divide both sides by -3 to isolate y


y+=+%28-4x%29%2F%28-3%29%2B%2810%29%2F%28-3%29 Break up the fraction on the right hand side


y+=+%284%2F3%29x-10%2F3 Reduce and simplify


The original equation 4x-3y=10 (standard form) is equivalent to y+=+%284%2F3%29x-10%2F3 (slope-intercept form)


The equation y+=+%284%2F3%29x-10%2F3 is in the form y=mx%2Bb where m=4%2F3 is the slope and b=-10%2F3 is the y intercept.





Now let's find the perpendicular line

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 4%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%284%2F3%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%283%2F4%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%2F4 Multiply the fractions.


So the perpendicular slope is -3%2F4



So now we know the slope of the unknown line is -3%2F4 (its the negative reciprocal of 4%2F3 from the line y=%284%2F3%29%2Ax-10%2F3). Also since the unknown line goes through (8,-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=%28-3%2F4%29%2A%28x-8%29 Plug in m=-3%2F4, x%5B1%5D=8, and y%5B1%5D=-3



y%2B3=%28-3%2F4%29%2Ax%2B%283%2F4%29%288%29 Distribute -3%2F4



y%2B3=%28-3%2F4%29%2Ax%2B24%2F4 Multiply



y=%28-3%2F4%29%2Ax%2B24%2F4-3Subtract -3 from both sides to isolate y

y=%28-3%2F4%29%2Ax%2B24%2F4-12%2F4 Make into equivalent fractions with equal denominators



y=%28-3%2F4%29%2Ax%2B12%2F4 Combine the fractions



y=%28-3%2F4%29%2Ax%2B3 Reduce any fractions

So the equation of the line that is perpendicular to y=%284%2F3%29%2Ax-10%2F3 and goes through (8,-3) is y=%28-3%2F4%29%2Ax%2B3


So here are the graphs of the equations y=%284%2F3%29%2Ax-10%2F3 and y=%28-3%2F4%29%2Ax%2B3




graph of the given equation y=%284%2F3%29%2Ax-10%2F3 (red) and graph of the line y=%28-3%2F4%29%2Ax%2B3(green) that is perpendicular to the given graph and goes through (8,-3)

Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
4x-3y=10
we find the slope by converting the equation into the form: y=mx+b where m=slope
subtract 4x from each side. -3y=-4x+10
divide -3 into each side. y=(4/3)x-10/3
so m=4/3
by definition the slope of the line perpendicular to this line is the negative reciprocal of its slope. So m=-1%2Fm%5B1%5D=-3%2F4.
we use the point-slope form to find the formula of the line.
y-y%5B1%5D=m%28x-x%5B1%5D%29+
for (8,-3): y-(-3)=-3/4(x-8)
y-(-3)=-3x/4+3/4*8
y+3=-.75x+6
subtract 3 from each side. y=-.75x+3
Ed
graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1.25x-10%2F3%2C-.75x%2B3%29