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Math A |
Solving Systems of Equations Algebraically -
Using Addition or Subtraction |
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Solve this system of
equations using the addition or subtraction method. Check.
x - 2y = 14
x + 3y = 9
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Simultaneous equations got you baffled? Relax!
You can do it!
Think of
the adding or subtracting method as simply "eliminating"
one of the variables to make your life easier. |
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Let's look at three examples using the
"addition" or "subtraction" method for simultaneous equations:
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1. Solve this system of equations and check: |
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x - 2y = 14
x + 3y = 9 |
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a. First, be sure that
the variables are "lined up" under one another. In
this problem, they are already "lined up". |
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x - 2y = 14
x + 3y = 9 |
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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. |
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x
- 2y = 14
x + 3y = 9 |
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c. Now,
subtract to
eliminate the "x" variable.
(Remember: when you subtract signed numbers, you
change the signs and follow the rules for adding signed
numbers.) |
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x - 2y = 14
-x
-
3y = -
9
- 5y = 5 |
| d.
Solve this simple equation. |
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-5y = 5
y = -1 |
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e. Plug "y
= -1" into either of the ORIGINAL equations to get the
value for "x". |
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x - 2y = 14
x - 2(-1) = 14
x + 2 = 14
x = 12 |
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f.
Check:
substitute x = 12 and y = -1 into BOTH ORIGINAL equations.
If these answers are correct, BOTH equations will be TRUE! |
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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!) |
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There is no stopping us now!
Let's try a harder problem.... |
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2. Solve this system of equations and check: |
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4x + 3y = -1
5x + 4y = 1 |
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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! |
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4x + 3y = -1
5x + 4y = 1 |
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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:
* multiply the two differing coefficients to obtain
the new coefficient (unless a small number can be found.)
* multiply EVERY element in the equation by your
adjustment number. |
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4(4x
+ 3y = -1)
3(5x + 4y = 1)
16x + 12y = -4
15x + 12y = 3
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c. Now,
subtract to
eliminate the "y" variable.
(Remember: when you subtract signed numbers, you
change the signs and follow the rules for adding signed
numbers.) |
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16x + 12y = -4
-15x
-
12y = -
3
x
= - 7 |
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d. Plug "x
= -7" into either of the ORIGINAL equations to get the
value for "y". |
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5x + 4y = 1
5(-7) + 4y = 1
-35 + 4y = 1
4y = 36
y = 9 |
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e.
Check:
substitute x = -7 and y = 9 into BOTH ORIGINAL equations.
If these answers are correct, BOTH equations will be TRUE! |
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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!)
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Let's finish with an addition method
problem:
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3. Solve this system of equations and check: |
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4x - y = 10
2x = 12 - 3y |
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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. |
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4x - y = 10
2x + 3y = 12 |
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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.) |
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3(4x
- y = 10)
2x + 3y = 12
12x - 3y = 30
2x + 3y = 12 |
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c. Now,
add to eliminate
the "y" variable.
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12x - 3y = 30
2x + 3y = 12
14x = 42 |
| d.
Solve this simple equation. |
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14x = 42
x = 3 |
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e. Plug "x
= 3" into either of the ORIGINAL equations to get the
value for "y". |
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4x - y = 10
4(3) - y = 10
12 - y = 10
-y = -2
y = 2 |
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f.
Check:
substitute x = 3 and y = 2 into BOTH ORIGINAL equations.
If these answers are correct, BOTH equations will be TRUE! |
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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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