Topic: EQUATIONS
Simple Equations
Fractional Equations
Guidelines
• Equations must be balanced. You
must respect the laws of equations.
• The goals is to bring variable on
the left and number to the right.
• Coefficient (or number) of variable
+1.
For example, 1x = 5; just put x = 5
Examples
1. #1) Solve: x = 5 + 2
x=7
2. x = 5 (2 + 7) – ( 7 – 3)
x = 10 + 35 – 7 + 3
• Remember a minus before a
bracket changes the sign of
everything in the bracket
x = 41
Guidelines con’t
• Whatever you do to one side, you
must do to another. If you add 5 to
one side, you must add 5 to the other
side.
• If you have a number near the
variable, always divide it by that
number. For example 2x = 10; divide
both by 2. x = 5
• If -5x = 10; divide by -5; x = -2
• When it changes signs, it changes
signs
Example
• 4x + 1 = 13
4x = 13 - 1 (the 1 changed sides
so it changes signs)
4x = 12 (divide both sides by 4)
x=3
Verification
• If you want to guarantee that you
have the right answer you should
verify. To verify: replace the number
into the letter in the question.
• 4x + 1 = 13 (Original question) & x=3
4 (3) + 1 = 13
12+1=13
13=13 This is true; you have the right
answer.
Guideline & Example
•
•
•
•
3x – 5 = 10x + 10
-3x - 10x = 10 + 5
-13x = 15
x = -1.15
Another Example – Long
Version
•
•
•
•
3x – 5 = 8x + 15
3x – 8x – 5 + 5 = 8x – 8x + 15 + 5
-5x = 20
x = -4
Same example – Short
Version
•
•
•
•
3x – 5 = 8x + 15
-5x = 20
x = -4
We will continue with the short
version ;)
More examples
•
•
•
•
•
•
•
4x + 7 = 2x – 11
2x = -18
x=-9
Verify!
4 (-9) + 7 = 2 (-9) – 11
-36 + 7 = -18 – 11
- 29 = -29 (You have the right answer)
Another Example
• 9x – 5 = 2x + 4
7x = 9
x = 1.29
You can still verify this!
9 (1.29) – 5 = 2 (1.29) + 4
6.61 = 6.58 (rounding error…
close enough!)
More Examples
• 3x + 5 = 6x + 25
• x = - 6.67
• 4x + 2 = 8x – 31
• X = 8.25 --- are you verifying?)
Reminders
• Distributive property
• 5(3x + 2) means you multiply 5
by everything in the bracket
• 15x + 10
• A minus sign before the bracket
changes the sign of everything
in the bracket
Now to add some fun –
and have them longer
• 3 (5x-7) – (2x+8) = 2 (3x-1)
• 15x – 21 – 2x – 8 = 6x – 2
• CLEAN IT UP BEFORE MOVING
NUMBERS OR LETTERS
• 13x – 29 = 6x – 2
• 7x = 27
• x = 3.86
• Verify!
Verify
• 3 (5x-7) – (2x+8) = 2 (3x-1)
• 3 (5(3.86) – 7) – (2(3.86) - 8 = 2
(3(3.86) – 1)
• 3 (12.3) – 7.72 – 8 = 2 (10.58)
• 21.18 = 21.16 (rounding error –
close enough!)
Quiz # 3 Equations
• 1)
• 2)
• 3)
• 4)
• 5)
5x
2x
9x
3x
6x
–
–
–
–
–
7
5
2
7
1
=
=
=
=
=
3x + 7
4x + 7
- 40 + 5x
8x + 20
8x + 20
Quiz #3 Con’t
• 6) 2 (3x – 7) = 5 (3x – 1)
• 7) 7(4x + 1) – 5 (3x + 5) =
8x – (3x + 2)
• 8) 2x – (3x + 1)
• 9) 2x + 5
• 10) – 5 – 5
Quiz #3 Equation
Solutions
•
•
•
•
•
•
•
•
•
•
1) 7 (1a)
2) – 6 (1b)
3) -9.5 (1i)
4) – 5.4 (1o)
5) -10.5
6) -1 (1u)
7) 2 (1w)
8) –x – 1 (minus before a bracket!)
9) 2x + 5 (don’t mix apples & oranges!)
10) – 10
Fractional Equations
• Once again we want to get rid
of the fraction
• Find the LCD (Lowest Common
Denominator)
• Multiply every term to get the
LCD.
Example
1.
x – 2 = 11
5 3 15
LCD of 5, 3 and 15 is
LCD is 15
3x – 10 = 11
3x = 21
X = 7 and then yes… VERIFY
Verify
1. x – 2 = 11
5 3 15
7/5 – 2/3 = 11 / 15
0.73 = 0.73 it works!
Another example
• 2x – 11 = 3x – 5
7 14 28 7
8x – 22 = 3x – 20
5x = 2
x = 0.4
Another Example
• 3 – 11x = 5 + 5x
8 12
24 6
9 – 22x = 5 + 20x
-42x = -4
X = 0.1
Final Example (Hard)
• 3x – 2 – 11x + 8x = 3 – 7x + 1
5
3 30
15 2 15 30
18x – 20 – 11x + 16x = 45 – 14x +1
37x = 66
X = 1.78