Transcript PowerPoint Presentation 13: Algebra

PRESENTATION 13
Simple Equations
EQUATIONS
• An equation is a mathematical
statement of equality between two
or more quantities
•
It always contains an equal sign
• A formula is a particular type of
equation that states a
mathematical rule
WRITING EQUATIONS
•
The following examples illustrate writing
equations from given word statements
•
A number plus 20 equals 35:
•
Let n = the number
The equation would become: n + 20 = 35
•
Four times a number equals 40:
•
•
Let x = the number
Four times the number would then be 4x
The equation is now: 4x = 40
SUBTRACTION PRINCIPLE OF
EQUALITY
• The subtraction principle of
equality states:
•
•
If the same number is subtracted from
both sides of an equation, the sides
remain equal
The equation remains balanced
SUBTRACTION PRINCIPLE OF
EQUALITY
• Procedure for solving an equation
in which a number is added to the
unknown:
•
Subtract the number that is added to the
unknown from both sides of the
equation
SUBTRACTION PRINCIPLE OF
EQUALITY
• Example:
•
Solve x + 5 = 12 for x:
In the equation, the number 5 is added
to x so subtract 5 from both sides to
solve for x
x + 5 = 12
– 5 –5
x = 7
ADDITION PRINCIPLE OF
EQUALITY
• The addition principle of equality
states:
•
•
If the same number is added to both
sides of an equation, the sides remain
equal
The equation remains balanced
ADDITION PRINCIPLE OF
EQUALITY
• Procedure for solving an equation
in which a number is subtracted
from the unknown
•
•
Add the number, which is subtracted
from the unknown, to both sides of an
equation
The equation maintains its balance
ADDITION PRINCIPLE OF
EQUALITY
• Example: Solve for y:
•
y – 7 = 10
In the equation, the number 7 is
subtracted from y, so add 7 to both
sides
y – 7 = 10
+ 7 +7
y = 17
DIVISION PRINCIPLE OF
EQUALITY
• The division principle of equality
states:
•
•
If both sides of an equation are divided
by the same number, the sides remain
equal
The equation remains balanced
DIVISION PRINCIPLE OF
EQUALITY
• Procedure for solving an equation
in which the unknown is multiplied
by a number:
•
•
Divide both sides of the equation by the
number that multiplies the unknown
The equation maintains its balance
DIVISION PRINCIPLE OF
EQUALITY
• Example: Solve for x: 6x = 30
•
In the equation, x is multiplied by 6, so
divide both sides by 6
6x 30

6
6
x=5
MULTIPLICATION PRINCIPLE OF
EQUALITY
• The multiplication principle of
equality states:
•
•
If both sides of an equation are
multiplied by the same number, the
sides remain equal
The equation remains balanced
MULTIPLICATION PRINCIPLE OF
EQUALITY
• Procedure for solving an equation
in which the unknown is divided by
a number:
•
•
Multiply both sides of the equation by
the number that divides the unknown
Equation maintains in balance
MULTIPLICATION PRINCIPLE OF
EQUALITY
• Example: Solve for y:
•
y
3
5
In the equation, y is divided by 3, so
multiply both sides by 3
y
(3 ) 5(3)
3
y = 15
ROOT PRINCIPLE OF EQUALITY
• The root principle of equality
states:
•
•
If the same root of both sides of an
equation is taken, the sides remain
equal
The equation remains balanced
ROOT PRINCIPLE OF EQUALITY
• Procedure for solving an equation
in which the unknown is raised to a
power:
•
•
Extract the root of both sides of the
equation that leaves the unknown with
an exponent of 1
Equation maintains in balance
ROOT PRINCIPLE OF EQUALITY
• Example: Solve for x:
x
•
2

2
x
= 25
25
In the equation, x is squared, so to solve
the equation, extract the square root of
both sides
x=5
POWER PRINCIPLE OF EQUALITY
• The power principle of equality
states:
•
•
If both sides of an equation are raised to
the same power, the sides remain equal
The equation remains balanced
POWER PRINCIPLE OF EQUALITY
• Procedure for solving an equation
which contains a root of the
unknown:
•
•
Raise both sides of the equation to the
power that leaves the unknown with an
exponent of 1
Equation maintains in balance
POWER PRINCIPLE OF EQUALITY
• Example: Solve for y:
•
•
y 3
In the equation, y is expressed as a
square root, so to solve the equation,
square both sides
(√y)2 = (3)2
y=9
PRACTICAL PROBLEMS
• A company’s profit for the second
half year is $150,000 greater than
the profit for the first half year
• The total annual profit is $850,000
• What is the profit for the first half
year and the second half of the
year?
PRACTICAL PROBLEMS
• Let P equal the profit for the first half year
• Then P + $150,000 is the profit for the
second half year
• The sum is $850,000
• Set up an equation:
•
P + P + $150,000 = $850,000
Sum like terms:
2P + $150,000 = $850,000
PRACTICAL PROBLEMS
• Use the subtraction principle of
equality and subtract $150,000 from
both sides:
2P + $150,000 – $150,000 = $850,000 – $150,000
2P = $700,000
• Use the division principle of equality
and divide both sides by 2
2P ÷ 2 = $700,000 ÷ 2
P = $350,000
PRACTICAL PROBLEMS
• The profit for the first half year is
$350,000
• The profit for the second half year
is $350,000 + $150,000 =
$500,000
• Check: Total profit is $350,000 +
$500,000, which equals $850,000