|
Solve this system of
equations using the addition or subtraction method. Check.
x - 2y = 14
x + 3y = 9
 |
Simultaneous equations got you baffled? Relax!
You can do it!
Think of
the adding or subtracting method as temporarily "eliminating"
one of the variables to make your life easier. |
|
Systems of Equations may also be referred to as
"simultaneous equations".
"Simultaneous" means being solved "at the same time".
Let's look at three examples using the
"addition" or "subtraction" method for systems of equations:
1. Solve this system of equations
and check: |
|
x - 2y = 14
x + 3y = 9 |
|
a. First, be sure that
the variables are "lined up" under one another. In
this problem, they are already "lined up". |
|
x - 2y = 14
x + 3y = 9 |
|
b. Decide
which variable ("x" or "y") will be easier to eliminate.
In order to eliminate a variable, the numbers in front of
them (the coefficients) must be the same or negatives of
one another.
Looks like "x" is the easier variable to eliminate in this
problem since the x's already have the same
coefficients. |
|
x
- 2y = 14
x + 3y = 9 |
|
c. Now,
in this problem we need to
subtract to
eliminate the "x" variable.
Subtract ALL of the sets of lined up terms.
(Remember: when you subtract signed numbers, you
change the signs and follow the rules for adding signed
numbers.) |
|
x - 2y = 14
-x
-
3y = -
9
- 5y = 5 |
| d.
Solve this simple equation. |
|
-5y = 5
y =
-1 |
|
e. Plug "y
= -1" into either of the ORIGINAL equations to get the
value for "x". |
|
x - 2y = 14
x - 2(-1) = 14
x + 2 = 14
x = 12 |
|
f.
Check:
substitute x = 12 and y = -1 into BOTH ORIGINAL equations.
If these answers are correct, BOTH equations will be TRUE! |
|
x - 2y = 14
12
- 2(-1)
= 14
12 + 2 = 14
14 = 14
(check!)
x + 3y = 9
12
+ 3(-1)
= 9
12 - 3 = 9
9 = 9
(check!) |
|
|
 |
There is no stopping us now!
Let's try a harder problem.... |
2. Solve this system of equations
and check: |
|
4x + 3y = -1
5x + 4y = 1 |
|
a. You can probably see
the dilemma with this problem right away. Neither of
the variables have the same (or negative) coefficients to
eliminate. Yeek! |
|
4x + 3y = -1
5x + 4y = 1 |
|
b.
In this type
of situation, we must MAKE the coefficients the same (or
negatives) by multiplication. You can
MAKE either the "x" or the "y"
coefficients the same. Pick the easier numbers.
In this problem, the "y" variables will be
changed to the same coefficient by multiplying the top
equation by 4 and the bottom equation by 3.
Remember:
* you can multiply the two differing coefficients to obtain
the new coefficient if you cannot think of another smaller
value that will work.
* multiply EVERY element in each equation by your
adjustment numbers. |
|
4(4x
+ 3y = -1)
3(5x + 4y = 1)
16x + 12y = -4
15x + 12y = 3
|
|
c. Now,
in this problem we need to
subtract to
eliminate the "y" variable.
(Remember:
when you subtract signed numbers, you change the signs
and follow the rules for adding signed numbers.) |
|
16x + 12y = -4
-15x
-
12y = -
3
x
= - 7 |
|
d.
Plug "x = -7" into either of the ORIGINAL
equations to get the value for "y". |
|
5x + 4y = 1
5(-7) + 4y = 1
-35 + 4y = 1
4y = 36
y = 9 |
|
e.
Check: substitute x = -7 and y = 9 into BOTH ORIGINAL equations.
If these answers are correct, BOTH equations will be TRUE! |
|
4x + 3y = -1
4(-7) +3(9) = -1
-28 + 27 = -1
-1 = -1
(check!)
5x + 4y = 1
5(-7) + 4(9) = 1
-35 + 36 = 1
1 = 1
(check!)
|
|
Let's finish with an addition method
problem:
|
3. Solve this system of equations and check: |
|
4x - y = 10
2x = 12 - 3y |
|
a. First, be sure that
the variables are "lined up" under one another. The
second equation was rearranged so that the variables would
"line up" with those in the first equation. |
|
4x - y = 10
2x + 3y = 12 |
|
b. Decide
which variable ("x" or "y") will be easier to eliminate.
In this problem, we must MAKE EITHER the "x" or the "y"
coefficients the same. The "y" variable is being
used here. Multiplying by 3 will give the "y"
variables negative coefficients. (Yes, -3 could also
have been used.) |
|
3(4x
- y = 10)
2x + 3y = 12
12x - 3y = 30
2x + 3y = 12 |
|
c. Now,
add to eliminate
the "y" variable.
|
|
12x - 3y = 30
2x + 3y = 12
14x = 42 |
| d.
Solve this simple equation. |
|
14x = 42
x = 3 |
|
e. Plug "x
= 3" into either of the ORIGINAL equations to get the
value for "y". |
|
4x - y = 10
4(3) -
y = 10
12 - y = 10
-y = -2
y = 2 |
|
f.
Check: substitute x = 3 and y = 2 into BOTH ORIGINAL equations.
If these answers are correct, BOTH equations will be TRUE! |
|
4x - y = 10
4(3) -
2 = 10
12 - 2 = 10
10 = 10
(check!)
2x = 12 - 3y
2(3) = 12 - 3(2)
6 = 12 - 6
6 = 6
(check!) |
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Grab you
Graphing Calculator:
Even though you
are doing an "algebraic solution" you can
still use your graphing calculator as a
CHECK to be sure you have the correct
answer. Click the calculator at
the right for directions on using the
TI-83+/84+ graphing calculator to solve
systems of equations.
Click calculator. |
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