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6/5/2015
SOLVING EQUATIONS
For instance, if 2x + 3 = 17,
then 7 is the only value of x
that will make this a true
sentence.
•
The single most important
skill in algebra
•
Goal: Find the one value
of the variable that
makes the sentence true.
•
We can solve equations by
doing the OPPOSITE of what
has been done to the
variable in the problem.
If a problem says +, you
subtract.
If a problem has
multiplication, you divide.
•
•
So x = 7.
By doing the opposite, we keep
the sides of the equation
balanced.
As long as you do the SAME
thing to both sides of an
equation, it will remain balanced.
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6/5/2015
7x – 13 = 50
7x – 13 = 50
+13 +13
7x
= 63
7x – 13 = 50
+13 +13
7x
= 63
7
7
x
= 9
-5x + 7 = 82
-5x + 7 = 82
-7 -7
-5x
= 75
-5x + 7 = 82
-7 -7
-5x
= 75
-5
-5
x
= -15
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6/5/2015
4x + 25 = 13
4x + 25 = 13
- 25 -25
4x
= -12
4
4
x
= -3
4x + 25 = 13
- 25 -25
4x
= -12
When you solve equations,
you also do the opposite of
the order of operations.
•
Add/subtract first
•
Then divide/multiply
Solve these equations:
Solve these equations:
4a + 11 = 59
4a + 11 = 59
Î a = 12
-2b + 13 = 5
-2b + 13 = 5
Î b=4
5c – 72 = 98
5c – 72 = 98
Î c = 34
-4d + 11 = 47
-4d + 11 = 47
Î d = -9
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Solve these equations:
Solve these equations:
5x – 18 = 40
5x – 18 = 40 Îx = 58/5 or 11.6
2y + 73 = 54
-3z + 5 = 1
2y + 73 = 54 Îy = -19/2 or -9.5
_
4
-3z + 5 = 1 Îz = /3 or 1.3
Some equations are even
easier.
Some equations are even
easier.
n + 4 = 13
n + 4 = 13
Just subtract 4 … x = 9
5x = 35
5x = 35
Just divide by 5 … x = 7
Solve
17 = 3x – 7
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Solve
Solve
17 = 3x – 7
+7
+7
24 = 3x
17 = 3x – 7
+7
+7
24 = 3x
3 3
8= x
Solve
Solve
19 – 2x = 104
19 – 2x = 104
-19
-19
-2x = 85
Solve
19 – 2x = 104
-19
-19
-2x = 85
-2
-2
x = -85/2 or -42.5
If you know the basic
steps, you can quickly
do equations with
more difficult numbers
using a calculator.
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6/5/2015
12x + 1794 = 2127
12x + 1794 = 2127
963 – 25x = 704
963 – 25x = 704
n
+ 13 = 22
7
What about this?
n
+ 13 = 22
7
Fractions mean division, so
to cancel, we’ll subtract 13
and then multiply by 7.
Î n = 63
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Things that can complicate
solving equations …
Parentheses
• Use distributive property
first.
Like terms
• Combine them first.
3(2x – 5) = 27
3(2x – 5) = 27
3(2x – 5) = 27
6x – 15 = 27
6x
= 42
x
= 7
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-7(2x – 11) = 98
-7(2x – 11) = 98
-14x + 77 = 98
-14x
= 21
x
= -3/2 or -1.5
4p + 3 – 2p + 7 + 5p + 2 = 17
4p + 3 – 2p + 7 + 5p + 2 = 17
7p + 12 = 17
7p
=5
p
= 5/7
5(3x + 5) – 3(2x – 1) = 145
5(3x + 5) – 3(2x – 1) = 145
15x + 25 – 6x + 3 = 145
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5(3x + 5) – 3(2x – 1) = 145
15x + 25 – 6x + 3 = 145
9x + 28 = 145
The goal is always to
simplify.
Make the problem look like the
easy ones we know how to
solve.
7x – 15 = 2x + 70
5(3x + 5) – 3(2x – 1) = 145
15x + 25 – 6x + 3 = 145
9x + 28 = 145
9x = 117
x = 13
Variable on Both Sides
• Find the smaller number of
the variable, and
subtract that on both
sides.
• Solve the remaining
problem.
7x – 15 = 2x + 70
-2x
-2x
5x – 15 =
70
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7x – 15 = 2x + 70
-2x
-2x
5x – 15 =
70
5x
=
85
x
=
17
5x + 13 = 7x + 40
5x + 13 = 7x + 40
-5x
-5x
13 = 2x + 40
5x + 13 = 7x + 40
-5x
-5x
13 = 2x + 40
x = -27/2 or -13.5
3(2x + 7) = 3x + 4 + x + 9
3(2x + 7) = 3x + 4 + x + 9
6x + 21 = 4x + 13
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3(2x + 7) = 3x + 4 + x + 9
6x + 21 = 4x + 13
2x + 21 =
13
2x
=
-8
x
=
-4
Yesterday you saw this
equation:
3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x)
3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x
3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x)
3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x
21x – 2 = 20x – 28
3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x)
Solve it.
3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x)
3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x
21x – 2 = 20x – 28
-20x
-20x
x–2=
-28
+2
+2
x
=
-26
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Special equations
2(3x – 7) = 6x + 11
2(3x – 7) = 6x + 11
6x – 14 = 6x + 11
?????
10x – 15 = 5(2x – 3)
When variables cancel out…
•
10x – 15 = 5(2x – 3)
10x – 15 = 10x – 15
?????
10x – 15 = 5(2x – 3)
10x – 15 = 10x – 15
-15 = -15
•
• An equation with infinitely
many solutions can also be
called an IDENTITY.
If you have the exact
same thing on both sides
(like 8 = 8), the answer is
ALL REAL NUMBERS or
INFINITELY MANY
SOLUTIONS.
If there is something
different on the 2 sides
(like 5 = 7), there is NO
SOLUTION.
2(3x – 7) = 6x + 11
6x – 14 = 6x + 11
-14 = 11
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•
If there is something
different on the 2 sides
(like 5 = 7), there is NO
SOLUTION.
2(3x – 7) = 6x + 11
6x – 14 = 6x + 11
-14 = 11
13