Transcript Solving_Equations

Algebra 2
1.5
Pinkston
SAT Question
Which of the following is (are) true?
I.
(10 – 5) – 3 = 10 – (5 – 3) I. The associative
property doesn’t
II. (2 × 3) × 5 = 2 × (3 × 5)
work for
III. (2 + 3) + 5 = 2 + (3 + 5)
subtraction.
A. I only
II and III. The
B. II only
associative property
C. III only
does work for
D. I and III
multiplication and
addition.
E. II and III
Equations are solved by using inverse operations.
For first degree equations, the aim is to get the
variable on one side of the equation by using the
following two properties:
Addition Property of Equality
If a  b, then a  c  b  c.
Multiplication Property of Equality
If a  b, then a  c  b  c.
For second degree equations (quadratic
equations), the equation is set equal to zero and
solved by various methods that we will learn later
in the year…
Examples:
Linear equations
3x  4  13
4 4
3x  17
3 3
17
x
3
3x  4  13
3x  17
17
x
3
17 
Solution: x   
3
You may want to learn to skip the extra steps.
8x  6  2x  12  4x  5
6x  6  7  4x
4 x
4 x
10x  6  7
6 6 Keep equal sign
lined up.
10x  13
 13 
Solution: x   

 10 
8x  6  2x  12  4x  5
6x  6  7  4x
4 x
4 x
10x  6  7
6 6 You can skip
those steps
10x  13
 13 
Solution: x   

 10 
SOLVING RATIONAL
EQUATIONS
If there is one fraction on each side of the equation, it is a
proportion.
=
If there is more than one fraction on each side of the
equation, it is a rational equation.
=
To solve a proportion, we cross-multiply:
A
C
=
B
D
AD = BC
Example:
x 3

8 7
7x  24
7 x  24
7 x 24

7
7
24 

Solution:x 
 
7
Example:
Turn it into a
proportion
2
x7
3
2x 7

3 1
2x  21
Solution:
 21 
x 
2
How do we solve
rational equations?
We multiply the LCD,
And cancel to get rid of the fractions.
A simpler equation we’ll see.
How do we solve
rational equations?
x 2x 7


15 5 10
We multiply the LCD,
x (30) 2x (30) 7 (30)


15
5
10
And cancel to get rid of the fractions.
2
6
3
x (30) 2x (30) 7 (30)


15
5
10
A simpler equation we’ll see.
2x 12x  21
2x 12x  21
Then we finish solving the simpler equation:
10 x 21

10 10
 21 
Solution: x   

 10 
How do we solve
rational equations?
3
2
7


2a 5a 10
We multiply the LCD,
3 (10a) 2 (10a) 7 (10a)


2a
5a
10
And cancel to get rid of the fractions.
5
2
a
3 (10a) 2 (10a) 7 (10a)


2a
5a
10
A simpler equation we’ll see.
15  4  7a
15  4  7a
Then finish solving:
Solution:
11 7 a

7
7
11 
 a
7
Example:
1
3 1
 y 
4
2 2
y
3
1
Re-write   
4 2 2
1
y
32
12
 (4)  (4)  (4)
4
2
2
Keep equal sign
y  6  2
lined up.
6 6
Solution:
 y  4
y  4
1 1
SAT Question:
The symbol ⌂ represents on of the four
fundamental operations of arithmetic; b and c
are different integers; and b  0.
If b ⌂ c = c ⌂ b and b ⌂ 0 = b, then the symbol ⌂
must represent
This is commutative. It only
A. + only
works for adding and
multiplying.
B. × only
C. + and ×
This is identity for
D. addition only.
E. ÷
Get ready for a “Small Quiz”
to be written
on your grade sheet.
Quiz. Copy the problems and write the
answer.
Simplify:
1. 2(3x  5)
2. 7 x  5  9 x 11
3. 3  5 x  (12  4 x)
Put your grade paper on the front of
your row, quiz side down.