Transcript Review of Equations - David Michael Burrow

SOLVING ONE-VARIABLE
EQUATIONS
•
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.
By doing the opposite, we keep
the sides of the equation
balanced.
5x – 13 = 52
5x – 13 = 52
+13 +13
5x
= 65
5x – 13 = 52
+13 +13
5x
= 65
5
5
x
= 13
12x + 1794 = 2127
12x + 1794 = 2127
x = 27.75
963 – 25x = 704
963 – 25x = 704
x = 10.36
What about this?
𝑦
− 13 = 2
5
What about this?
𝑦
− 13 = 2
5
Fractions mean division, so
to cancel, we’ll add 13 and
then multiply by 5.
 n = 75
Things that can complicate
solving equations …
Parentheses
• Use distributive property
first.
Like terms
• Combine them first.
4(3x – 7) = 48
4(3x – 7) = 48
12x – 28 = 48
4(3x – 7) = 48
12x – 28 = 48
12x
= 76
x
= 6.333…
-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
5(3x + 5) – 3(2x – 1) = 145
15x + 25 – 6x + 3 = 145
9x + 28 = 145
5(3x + 5) – 3(2x – 1) = 145
15x + 25 – 6x + 3 = 145
9x + 28 = 145
9x = 117
x = 13
The goal is always to
simplify.
Make the problem look like the
easy ones we know how to
solve.
Variable on Both Sides
• Find the smaller number of
the variable, and
subtract that on both
sides.
• Solve the remaining
problem.
5x – 15 = 2x + 72
5x – 15 = 2x + 72
-2x
-2x
3x – 15 =
72
7x – 15 = 2x + 72
-2x
-2x
3x – 15 =
72
3x
=
87
x
=
29
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
3(2x + 7) = 3x + 4 + x + 9
6x + 21 = 4x + 13
2x + 21 =
13
2x
=
-8
x
=
-4
Special equations
2(3x – 7) = 6x + 11
10x – 15 = 5(2x – 3)
2(3x – 7) = 6x + 11
6x – 14 = 6x + 11
?????
10x – 15 = 5(2x – 3)
10x – 15 = 10x – 15
?????
When variables cancel out…
•
If you have the exact
same thing on both sides
(like 8 = 8), the answer is
ALL REAL NUMBERS or
INFINITELY MANY
SOLUTIONS.
10x – 15 = 5(2x – 3)
10x – 15 = 10x – 15
-15 = -15
• An equation with infinitely
many solutions can also be
called an IDENTITY.
•
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
•
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