SOLUTION: a. Find the equation of the straight line joinging the points (1,2) and (-3,5). b. Find the equation of the straight line passing through the point (1,2) and perpendicular to the

Algebra ->  College  -> Linear Algebra -> SOLUTION: a. Find the equation of the straight line joinging the points (1,2) and (-3,5). b. Find the equation of the straight line passing through the point (1,2) and perpendicular to the       Log On


   

Question 84243: a. Find the equation of the straight line joinging the points (1,2) and (-3,5).
b. Find the equation of the straight line passing through the point (1,2) and perpendicular to the above straight line.
c. What is the point of intersection fo the two lines?
d. Algebraically verify your answer to above part of the problem.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a.
Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (1,2) and (-3,5)


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 (1,2) and (x%5B2%5D,y%5B2%5D) is the second point (-3,5))


m=%285-2%29%2F%28-3-1%29 Plug in y%5B2%5D=5,y%5B1%5D=2,x%5B2%5D=-3,x%5B1%5D=1 (these are the coordinates of given points)


m=+3%2F-4 Subtract the terms in the numerator 5-2 to get 3. Subtract the terms in the denominator -3-1 to get -4




m=-3%2F4 Reduce



So the slope is

m=-3%2F4





------------------------------------------------


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-2=%28-3%2F4%29%28x-1%29 Plug in m=-3%2F4, x%5B1%5D=1, and y%5B1%5D=2 (these values are given)



y-2=%28-3%2F4%29x%2B%28-3%2F4%29%28-1%29 Distribute -3%2F4


y-2=%28-3%2F4%29x%2B3%2F4 Multiply -3%2F4 and -1 to get 3%2F4

y=%28-3%2F4%29x%2B3%2F4%2B2 Add 2 to both sides to isolate y


y=%28-3%2F4%29x%2B11%2F4 Combine like terms 3%2F4 and 2 to get 11%2F4 (note: if you need help with combining fractions, check out this solver)



------------------------------------------------------------------------------------------------------------

Answer:



So the equation of the line which goes through the points (1,2) and (-3,5) is:y=%28-3%2F4%29x%2B11%2F4


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-3%2F4 and the y-intercept is b=11%2F4


Notice if we graph the equation y=%28-3%2F4%29x%2B11%2F4 and plot the points (1,2) and (-3,5), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=%28-3%2F4%29x%2B11%2F4 through the points (1,2) and (-3,5)


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



b.
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 -3%2F4, 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%28-3%2F4%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%284%2F-3%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=4%2F3 Multiply the fractions.


So the perpendicular slope is 4%2F3



So now we know the slope of the unknown line is 4%2F3 (its the negative reciprocal of -3%2F4 from the line y=%28-3%2F4%29%2Ax%2B11%2F4). Also since the unknown line goes through (1,2), 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=%284%2F3%29%2A%28x-1%29 Plug in m=4%2F3, x%5B1%5D=1, and y%5B1%5D=2



y-2=%284%2F3%29%2Ax-%284%2F3%29%281%29 Distribute 4%2F3



y-2=%284%2F3%29%2Ax-4%2F3 Multiply



y=%284%2F3%29%2Ax-4%2F3%2B2Add 2 to both sides to isolate y

y=%284%2F3%29%2Ax-4%2F3%2B6%2F3 Make into equivalent fractions with equal denominators



y=%284%2F3%29%2Ax%2B2%2F3 Combine the fractions



y=%284%2F3%29%2Ax%2B2%2F3 Reduce any fractions

So the equation of the line that is perpendicular to y=%28-3%2F4%29%2Ax%2B11%2F4 and goes through (1,2) is y=%284%2F3%29%2Ax%2B2%2F3


So here are the graphs of the equations y=%28-3%2F4%29%2Ax%2B11%2F4 and y=%284%2F3%29%2Ax%2B2%2F3




graph of the given equation y=%28-3%2F4%29%2Ax%2B11%2F4 (red) and graph of the line y=%284%2F3%29%2Ax%2B2%2F3(green) that is perpendicular to the given graph and goes through (1,2)



c.
Point of intersection (1,2) (this is given and is clearly visible)
d.
%28-3%2F4%29x%2B11%2F4=%284%2F3%29x%2B2%2F3 Start with the given expression
%28-3%2F4%29x%2B11%2F4-2%2F3=%284%2F3%29x Subtract 2%2F3 from both sides
11%2F4-2%2F3=%284%2F3%29x%2B%283%2F4%29x Add %283%2F4%29x to both sides
25%2F12=%2825%2F12%29x Combine like terms
%2825%2F12%29%2812%2F25%29=x Multiply both sides by 12%2F25
x=1 Simplify
y=%284%2F3%291%2B2%2F3 Plug in x=1
y=4%2F3%2B2%2F3 Multiply
y=6%2F3 Add
y=2 Reduce
So the intersection is (1,2). This verifies our original answer.