SOLUTION: I am having problems with solving equations. The following are equations and it's methods that I don't seem to understand. Thank you for your assistance! Solve using the add

I am having problems with solving equations. The 
following are equations and it's methods that I don't 
seem to understand. Thank you for your assistance! 

Solve using the addition (elimination) method.

They are already in standard form

Ax + By = C
Dx + Ey = F

If they weren't in that form we'd have to get them 
that way, but they already are.

2x +  3y = -16
5x - 10y =  30 

Look at the coefficients of x.  They are 2 and 5.  
We want to cause new coefficients to occur where 
they are that will  be equal in absolute value but 
opposite in sign so they will cancel out.  The least 
common multiple of 2 and 5 is 10.  We can cause a 
10x to appear where the 2x is by multiplying the 
first equation through by 5.  We can cause a 10x to 
appear where the 5x is by multiplying the second 
equation through by 2.  But we will choose one of 
these multipliers to be negative (it doesn't matter 
which one) so that they will be opposite in sign and 
cancel.  So I will multiply the first equation by -5 
and the second equation by +2.

-5[2x +  3y = -16]
 2[5x - 10y =  30]

Multiplying through:

 -10x - 15y = 80
  10x - 20y = 60
 
Now we draw a line under them and add vertically

 -10x - 15y = 80
  10x - 20y = 60
 -----------------
       -35y = 140
          y = 140/(-35)
          y = -4
          
2x +  3y = -16
5x - 10y =  30 

Now look at the coefficients of y.  They are 3 and -10.
We want to cause new coefficients to occur where they 
are that will  be equal in absolute value but opposite 
in sign so they will cancel out.  The least common 
multiple of 3 and 10 is 30.  We can cause a 30y to 
appear where the 3y is by multiplying the first equation
through by 10.  We can cause a -30y to appear where the
-10y is by multiplying the second equation through by 3.
We don't need choose to them opposite in sign as we did
before because they already are opposite in sign.  So I
will multiply the first equation by 10 and the second 
equation by 3.

10[2x +  3y = -16]
 3[5x - 10y =  30]

Multiplying through:

  20x + 30y = -160
  15x - 30y =   90
 
Now we draw a line under them and add vertically

  20x + 30y = -160
  15x - 30y =   90
 -----------------
  35x       =  -70 
          x = -70/35
          y = -2

So the solution is (x,y) = (-4,-2)

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

Solve using the addition (elimination) method

2y - 8 = -2x - 8x
8x - 3y = 31 + y 

Let's first get them like in stanbdard form like this:

Ax + By = C
Dx + Ey = F

The first one:
  2y - 8 = -2x - 8x
  2y - 8 = -10x
10x + 2y = 8

Notice that we can divide this equation through by 2. 
This isn't absolutely necessary by since we can it will
make it easier so we divide through by 2 and get

  5x + y = 8

Now working with the second equation:

  8x - 3y = 31 + y 
  8x - 2y = 31

So our system in standard order

  5x +  y =  8
  8x - 2y = 31

We choose one of the equations to solve for one of the
letters.  The easiest choice we can make is to pick the
first equation to solve for y.  (Whenever possible choose
a letter that has a 1 or -1 coefficient, since that makes
it easier).

Solving the first equation for y

  5x + y = 8
       y = 8 - 5x

Now go to the OTHER equation

 8x - 2y = 31

Substitute 8 - 5x for y in that: 
       
 8x - 2(8 - 5x) = 31  
  8x - 16 + 10x = 31
            18x = 47 
              x = 47/18

Now we substitute 47/18 for x in the equation solved 
for y:

y = 8 - 5x
y = 8 - 5(47/18)
y = 8 - 235/18
y = 144/18 - 235/18
y = -91/18

So the solution is (x,y) = (47/18,-91/18)

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

Solve using the substitution method

x = 3y + 7
x = 2y - 1 

These are already solved for on letter, so we substitute
3y + 7 for x in the second equation:

     x = 2y - 1
3y + 7 = 2y - 1
     y = -8

Now substitute -8 for y in either one of the original
equations, say, the first:

x = 3y + 7
x = 3(-8) + 7
x = -24 + 7
x = -17

So the solution is (x, y) = (-17,-8) 

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

Solve using the substitution method

4x + 3y = 0
2x -  y = 0

Choose the second one to solve for y, since that's the
simpler.

2x - y = 0
    -y = -2x
     y = 2x

Substitute 2x for y in the OTHER equation

   4x + 3y = 0 
4x + 3(2x) = 0
   4x + 6x = 0
       10x = 0
         x = 0/10
         x = 0

Substitute 0 for x in the equation y = 2x

         y = 2x
         y = 2(0)
         y = 0

So the solution is (x,y) = (0,0) 

Edwin
AnlytcPhil@aol.com