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