Download Solving Equations with Fractions PPT

SOLVING EQUATIONS
W/ FRACTIONS USING
LCM!
SWBAT:
Solve Equations with Fractions.
Fractions? Oh No!
• Equations involving fractions can be quite difficult to work
with.
• There are easy ways to change the way a problem looks
by using Multiplication and LCM’s!
What is an LCM again?
• It’s ok if you forgot…
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Ex 1: Solve the equation
• Solve
3
7
a-1 = a+ 9
11
11
æ3
ö æ7
ö
11ç a-1÷ = ç a+ 9 ÷11
è 11
ø è 11
ø
3a-11= 7a+ 99
3a-11-3a = 7a+ 99 -3a
-11= 4a+ 99
-11- 99 = 4a+ 99 - 99
-110 = 4a
-27.5 = a
Since both fractions have a
denominator of 11, multiply each
side by 11!
Distribute the 11 to both sides!
Now our equation is something
more familiar!
We can solve this!
Lets undo operations!
Ex 1B: Solve the equation
• Solve
2x 4
=
3 3
æ 2x ö æ 4 ö
3ç ÷ = ç ÷ 3
è 3 ø è3ø
2x = 4
Since both fractions have a
denominator of 3, multiply each
side by 3!
Now our equation is something
more familiar!
We can solve this!
Lets undo operations!
x=2
Note: We could also have solved this problem by multiplying both sides by
3/2, the reciprocal of 2/3!
Pt. 1 Practice…
• 1.
• 3.
3
7
x +1 =
4
4
5
15
x=8
8
1) x=1; 2) x=14 3) x= -3 4) x= 3/7
2.
1
3
x + 3 = x -1
7
7
4.
9
1
2x + = 3+ x
4
4
Ex 2: Different Denominators
• Solve
2 1 1
- b+ =
3 9 18
æ 2 1ö æ 1 ö
18ç - b+ ÷ = ç ÷18
è 3 9 ø è 18 ø
-12b+ 2 =1
-12b = -1
-12b -1
=
-12 -12
b=
1
12
3,9,and 18 have a LCM of 18, so
multiply each side by 18!
Distribute the 18 to both sides!
Now our equation is something
more familiar!
We can solve this!
Ex 2B: Different Denominators
• Solve
-3 2
= x
4 5
æ 3ö æ2 ö
20 ç- ÷ = ç x÷ 20
è 4ø è5 ø
-15 = 8x
-15 8
= x
8 8
-15
=x
8
4 and 5 have a LCM of 20, so
multiply each side by 20!
Ex 2C: Different Denominators
• Solve
x+8 4+ x
=
3
4
æ x+8ö æ 4 + xö
12 ç
÷=ç
÷12
è 3 ø è 4 ø
4(x+8) = (4 + x)3
4x+32 =12 +3x
4x+32 -3x =12 +3x-3x
x+32 =12
x+32 -32 =12 -32
x = -20
3 and 4 have a LCM of 12, so
multiply each side by 12!
Pt. 2 Practice
• 1.
1
7
x-8 =
2
8
2.
5 3
= x
8 16
• 3.
5 3
1
+ x=
8 4
16
4.
2
3
4
x + =1- x
5
7
7
• 5.
x - 8 15
=
12
3
6.
x - 7 2x + 3
=
4
2
1) 71/4
2) 10/3
3) -3/4
4) 10/17 5) 68
6) -13/3
Ex 3: Distribution of FRACTIONS??
• Solve
2
( x+ 4) =16
3
2
(3) ( x+ 4) =16(3)
3
2 ( x+ 4) = 48
2x+8 = 48
2x+8-8 = 48-8
2x = 40
2x 40
=
2 2
x = 20
Distributing the 2/3 would give
us more fractions…
We can undo the fraction by
multiplying by 3, then only have
to distribute the 2!
Ex 3: OR……
• Solve
2
( x+ 4) =16
3
æ3ö2
æ3ö
ç ÷ ( x + 4) =16 ç ÷
è2ø 3
è2ø
1( x+ 4) = 24
x+ 4 = 24
x = 20
We can also undo the fraction by
multiplying by its reciprocal!
When multiplying a fraction by
its Reciprocal, we always will get
a product of 1!
Pt. 3 Practice…
1
( x+ 4) =10
4
2)
3) 1 10x-10 = 5x-17
(
)
4)
1)
5
1) 36,
2) 7,
3) 5,
4) 0
3 1
= (10 - x)
2 2
2
1
10x+
6
=
(
) (15x+12)
3
3
Ex 4: Variables in Denominators
• Solve
2 -8
=
3 x
æ 2 ö æ -8 ö
3xç ÷ = ç ÷ 3x
è 3ø è x ø
2x = -24
2x -24
=
2
2
x = -12
The Denominators 3 and x can
be eliminated by multiplying both
sides each one!
Ex 4B: Variables in Denominators?
• Solve
x, 3x and 9 have a LCM of 9x...
2 4 2
- =
Multiply by 9x! Clear those demoninators!
x 3x 9
æ2 4 ö æ2ö
9xç - ÷ = ç ÷ 9x
è x 3x ø è 9 ø
18-12 = 2x
6 = 2x
3= x
Ex 4C: Variables in Denominators
• Solve
9
-63
=
x -1 28
æ 9 ö æ -63 ö
( x-1) (28)çè ÷ø = çè ÷ø( x-1) (28)
x -1
28
(28) ( 9) = (-63) ( x-1)
252 = -63x+ 63
189 = -63x
-3 = x
The Denominators 28 and x – 1
can be eliminated by multiplying
both sides each one!
Pt. 4 Practice…
1)
4 -12
=
5
x
2)
3)
3 12
- =
4 2x
4)
5)
5
17
+6 =
x
x
6)
1) -15
2) 0
3) -8
4) 6
5) 2
6) 9
2
-26
=
x - 5 65
3
-18
=
x + 4 -5x - 30
x + 2 x -1
+
=5
3
6
Boom!
• Solve
2 - x x+ 3 -1
=
x
3x
3