SOLUTION: O is the midpoint of NP; NP = 3x + 4y , NO = 2x + 3y - 5, OP = 5x - y, FIND the lengths of NO,OP, and NP. May you please show work also. N-----O-----P

Algebra ->  Length-and-distance -> SOLUTION: O is the midpoint of NP; NP = 3x + 4y , NO = 2x + 3y - 5, OP = 5x - y, FIND the lengths of NO,OP, and NP. May you please show work also. N-----O-----P      Log On


   

Question 99730: O is the midpoint of NP; NP = 3x + 4y , NO = 2x + 3y - 5, OP = 5x - y, FIND the lengths of NO,OP, and NP. May you please show work also.
N-----O-----P
Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
A) 2x+3y-5=5x-y
subtract 5x and add y to each side: -3x+4y-5=0
add 5 to each side: -3x+4y=5
B) 3x+4y=2(2x+3y-5)
3x+4y=4x+6y-10
subtract 4 and 6y from each side: -x-2y=-10
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

-1%2Ax-2%2Ay=-10
-3%2Ax%2B4%2Ay=5

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get -1 and -3 to some equal number, we could try to get them to the LCM.

Since the LCM of -1 and -3 is 3, we need to multiply both sides of the top equation by -3 and multiply both sides of the bottom equation by 1 like this:

-3%2A%28-1%2Ax-2%2Ay%29=%28-10%29%2A-3 Multiply the top equation (both sides) by -3
1%2A%28-3%2Ax%2B4%2Ay%29=%285%29%2A1 Multiply the bottom equation (both sides) by 1


So after multiplying we get this:
3%2Ax%2B6%2Ay=30
-3%2Ax%2B4%2Ay=5

Notice how 3 and -3 add to zero (ie 3%2B-3=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%283%2Ax-3%2Ax%29%2B%286%2Ay%2B4%2Ay%29=30%2B5

%283-3%29%2Ax%2B%286%2B4%29y=30%2B5

cross%283%2B-3%29%2Ax%2B%286%2B4%29%2Ay=30%2B5 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

10%2Ay=35

y=35%2F10 Divide both sides by 10 to solve for y



y=7%2F2 Reduce


Now plug this answer into the top equation -1%2Ax-2%2Ay=-10 to solve for x

-1%2Ax-2%287%2F2%29=-10 Plug in y=7%2F2


-1%2Ax-14%2F2=-10 Multiply



-1%2Ax-7=-10 Reduce



-1%2Ax=-10%2B7 Subtract -7 from both sides

-1%2Ax=-3 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ax=%28-3%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.


x=3 Multiply the terms on the right side


So our answer is

x=3, y=7%2F2

which also looks like

(3, 7%2F2)

Notice if we graph the equations (if you need help with graphing, check out this solver)

-1%2Ax-2%2Ay=-10
-3%2Ax%2B4%2Ay=5

we get



graph of -1%2Ax-2%2Ay=-10 (red) -3%2Ax%2B4%2Ay=5 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (3,7%2F2). This verifies our answer.

Check:
NO) 6+10.5-5=11.5
OP) 15-3.5=11.5
NP) 9+14=23
11.5+11.5=23
Ed