SOLUTION: A line has the equation 2x – y = 3 a. Find the equation of a line passing through the point (1,-2) that is parallel to this line. b. Find the equation of a line passing th

Algebra ->  Graphs -> SOLUTION: A line has the equation 2x – y = 3 a. Find the equation of a line passing through the point (1,-2) that is parallel to this line. b. Find the equation of a line passing th      Log On


   

Question 85192: A line has the equation 2x – y = 3
a. Find the equation of a line passing through the point (1,-2)
that is parallel to this line.
b. Find the equation of a line passing through the point (1,-2) perpendicular to this line.
c. Graph all three lines on the same graph.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First convert 2x-y+=+3 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-1y=3 Start with the given equation


2x-1y-2x=3-2x Subtract 2x from both sides


-1y=-2x%2B3 Simplify


%28-1y%29%2F%28-1%29=%28-2x%2B3%29%2F%28-1%29 Divide both sides by -1 to isolate y


y+=+%28-2x%29%2F%28-1%29%2B%283%29%2F%28-1%29 Break up the fraction on the right hand side


y+=+2x-3 Reduce and simplify


The original equation 2x-1y=3 (standard form) is equivalent to y+=+2x-3 (slope-intercept form)


The equation y+=+2x-3 is in the form y=mx%2Bb where m=2 is the slope and b=-3 is the y intercept.






a.
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 (its from the slope of y=2%2Ax-3 which is also 2). 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%2B2=2%2A%28x-1%29 Plug in m=2, x%5B1%5D=1, and y%5B1%5D=-2



y%2B2=2%2Ax-%282%29%281%29 Distribute 2



y%2B2=2%2Ax-2 Multiply



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

y=2%2Ax-4 Combine like terms

So the equation of the line that is parallel to y=2%2Ax-3 and goes through (1,-2) is y=2%2Ax-4


So here are the graphs of the equations y=2%2Ax-3 and y=2%2Ax-4



graph of the given equation y=2%2Ax-3 (red) and graph of the line y=2%2Ax-4(green) that is parallel to the given graph and goes through (1,-2)






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 2, 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%2F1%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%281%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=-1%2F2 Multiply the fractions.


So the perpendicular slope is -1%2F2



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



y%2B2=%28-1%2F2%29%2Ax%2B%281%2F2%29%281%29 Distribute -1%2F2



y%2B2=%28-1%2F2%29%2Ax%2B1%2F2 Multiply



y=%28-1%2F2%29%2Ax%2B1%2F2-2Subtract -2 from both sides to isolate y

y=%28-1%2F2%29%2Ax%2B1%2F2-4%2F2 Make into equivalent fractions with equal denominators



y=%28-1%2F2%29%2Ax-3%2F2 Combine the fractions



y=%28-1%2F2%29%2Ax-3%2F2 Reduce any fractions

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


So here are the graphs of the equations y=2%2Ax-3 and y=%28-1%2F2%29%2Ax-3%2F2




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






c.
So here are all three graphs

+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+2x-3%2C+2x-4%2C%28-1%2F2%29x-3%2F2%29+ graph of 2x-3 (red), 2x-4 (green), and %28-1%2F2%29x-3%2F2%29 (black)