Transcript Solving One-Step Equations

Solving One-Step Equations
Section 2-1
Goals
Goal
• To solve one-step equations
in one variable.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
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Equivalent Equations
Addition Property of Equality
Subtraction Property of Equality
Isolate
Inverse Operations
Multiplication Property of Equality
Division Property of Equality
Linear Equations in One
Variable
A linear equation in one variable is an equation that
can be written in the form ax + b = c where a, b, and c
are real numbers and a  0.
3x + 5 = 25
4
 8a
7
The expressions are called the sides of the equation.
Solutions
The solution of a linear equation is the value or values
of the variable that make the equation a true statement.
The set of all solutions of an equation is called the
solution set. The solution satisfies the equation.
Example:
Determine if x = – 1 is a solution to the equation.
– 3(x – 3) = – 4x + 3 – 5x
– 3[(– 1) – 3] = – 4(– 1) + 3 – 5(– 1)
– 3(– 4) = 4 + 3 + 5
True. x = – 1 is a solution
12 = 12
Equivalent Equations
Linear equations are solved by writing a
series of steps that result in the equation
x = a number
One method for solving equations is to write a series
of equivalent equations.
Two or more equations that have precisely the same
solutions are called equivalent equations.
3+5=8
1+7=2+6
Equivalent Equations
Two equations are said to be equivalent if every solution
of either one is also a solution of the other.
( a)
x6 9
(b)
x  10  13
The two equations shown above are equivalent, because
the number 3 will satisfy both equations and 3 is the only
number that will satisfy either equation.
Inverse Operations
To find solutions, perform inverse operations until
you have isolated the variable. A variable is
isolated when it appears by itself on one side of an
equation, and not at all on the other side.
Isolated
Variable
x = a number
Inverse Operations
Add x.
Multiply by x.
Subtract x.
Divide by x.
An equation is like a balanced scale. To
keep the balance, you must perform the
same inverse operation on both sides.
Inverse Operations
Isolate a variable by using inverse operations
which "undo" operations on the variable.
An equation is like a balanced scale. To keep the
balance, perform the same operation on both sides.
Inverse Operations
Operation
Addition
Subtraction
Inverse Operation
Subtraction
Addition
Inverse Operations
Solving an equation that contains multiplication
or division is similar to solving an equation that
contains addition or subtraction. Use inverse
operations to undo the operations on the variable.
Inverse Operations
Operation
Inverse Operation
Multiplication
Division
Division
Multiplication
Addition Property Of Equality
The Addition Property of Equality states that for
real numbers a, b, and c,
if a = b, then a + c = b + c.
y - 6 = 11
y - 6 (+6) = 11 (+6)
We need to find
the value of y.
Adding (+6) to both sides of the equation
will maintain the balance of the equation.
y = 17
Solution to
the equation.
+6
+6
Left side
Right side
Using the Addition Property
of Equality
Example:
Solve the linear equation x  9 = 22.
Step 1: Isolate the variable x on the left side of the equation.
x  9 + 9 = 22 + 9
Add 9 to both sides of the equation.
Step 2: Simplify the left and right sides of the equation.
x = 31
Apply the Additive Inverse Property.
Step 3: Check to verify the solution.
x  9 = 22
31  9 = 22
22 = 22 
Properties of Equality
WORDS
NUMBERS
ALGEBRA
Addition Property of Equality
You can add the same number to both
sides of an equation, and the statement
will still be true.
3=3
3+2=3+2
5=5
a=b
a+c=b+c
Properties of Equality
WORDS
NUMBERS
ALGEBRA
Subtraction Property of Equality
You can subtract the same number from
both sides of an equation, and the
statement will still be true.
7=7
7–5=7–5
2=2
a=b
a–c=b–c
Writing Math
Solution sets are written in set notation using
braces, { }. Solutions may be given in set
notation, or they may be given in the form x = 14.
Example
Solve the equation.
y – 8 = 24
+8 +8
y = 32
Since 8 is subtracted from y,
add 8 to both sides to undo
the subtraction.
The solution set is {32}.
Check
y – 8 = 24
32 – 8 24
24 24 
To check your solution, substitute
32 for y in the original
equation.
Example
Solve the equation.
4.2 = t + 1.8
–1.8
–1.8
Since 1.8 is added to t, subtract 1.8
from both sides to undo the
addition.
2.4 = t
Check
The solution set is {2.4}.
4.2 = t + 1.8
4.2 2.4 + 1.8
4.2 4.2 
To check your solution, substitute
2.4 for t in the original equation.
Your Turn:
Solve the equation. Check your answer.
n – 3.2 = 5.6
+ 3.2 + 3.2
Since 3.2 is subtracted from n, add
3.2 to both sides to undo the
subtraction.
n = 8.8
Check
n – 3.2 = 5.6
8.8 – 3.2 5.6
5.6 5.6
The solution set is {8.8}.

To check your solution, substitute
8.8 for n in the original
equation.
Your Turn:
Solve the equation. Check your answer.
–6 = k – 6
+6
+6
Since 6 is subtracted from k,
add 6 to both sides to undo
the subtraction.
0=k
Check
The solution set is {0}.
–6 = k – 6
–6 0 – 6
–6 –6 
To check your solution, substitute 0
for k in the original equation.
Your Turn:
Solve the equation. Check your answer.
6 + t = 14
–6
–6
Since 6 is added to t, subtract 6 from
both sides to undo the addition.
t= 8
Check
6 + t = 14
6 + 8 14
14 14 
The solution set is {8}.
To check your solution, substitute
8 for t in the original equation.
Multiplication Property of
Equality
The Multiplication Property of Equality states that
for real numbers a, b, and c, where c  0,
if a = b, then ac = bc.
We need to find
the value of x.
1
x  4
7
7 1
7
 x  4
1 7
1
x = 28
Solution to
the equation.
Multiplying both sides of the equation by 7
1
will maintain the balance of the equation.
×7
×7
Left side
Right side
Using the Multiplication
Property of Equality
Example:
Solve the linear equation 3x = 81
Step 1: Get the coefficient of the variable x to be 1.
1
1
1
(3x) = (81) Multiply each side of the equation by 3 .
3
3
Step 2: Simplify the left and right sides of the equation.
x = 27
Apply the Multiplicative Inverse Property.
Step 3: Check to verify the solution.
3x = 81
3(27) = 81
81 = 81 
Properties of Equality
WORDS
NUMBERS
ALGEBRA
Multiplication Property of Equality
You can multiply both sides of an
equation by the same number, and the
statement will still be true.
6=6
6(3) = 6(3)
18 = 18
a=b
ac = bc
Properties of Equality
WORDS
Division Property of Equality
You can divide both sides of an equation by the
same nonzero number, and the statement will still
be true.
NUMBERS
8=8
8
8
=
4
4
2 =2
ALGEBRA
a=b
(c ≠ 0)
a = a
c
c
Example:
Solve the equation. Check your answer.
Since j is divided by 3, multiply from
both sides by 3 to undo the division.
–24 = j
The solution set is {–24}.
Check
To check your solution, substitute –24 for
j in the original equation.
–8
–8 
Example:
Solve the equation. Check your answer.
–4.8 = –6v
0.8 = v
Check
The solution set is {0.8}.
–4.8 = –6v
–4.8 –6(0.8)
–4.8
Since v is multiplied by –6, divide
both sides by –6 to undo the
multiplication.
–4.8 
To check your solution, substitute
0.8 for v in the original
equation.
Your Turn:
Solve each equation. Check your answer.
Since p is divided by 5, multiply both
sides by 5 to undo the division.
p = 50
Check
The solution set is {50}.
To check your solution, substitute
50 for p in the original
equation.
10 10 
Your Turn:
Solve each equation. Check your answer.
0.5y = –10
y = –20
Check
0.5y = –10
0.5(–20) –10
–10 –10 
Since y is multiplied by 0.5, divide both
sides by 0.5 to undo the
multiplication.
The solution set is {–20}.
To check your solution, substitute
–20 for y in the original
equation.
Your Turn:
Solve each equation. Check your answer.
Since c is divided by 8, multiply both
sides by 8 to undo the division.
c = 56
Check
The solution set is {56}.
To check your solution, substitute
56 for c in the original
equation.
7
7 
Solving Equations
When solving equations, you will sometimes find it
easier to add an opposite to both sides instead of
subtracting or to multiply by a reciprocal instead of
dividing. This is often true when an equation contains
negative numbers or fractions.
Example:
Solve each equation.
The reciprocal of
is
w is multiplied by
both sides by
. Since
multiply
.
The solution set is {–24}.
Example:
Solve each equation.
Since p is added to
, add
to both sides to undo the
subtraction.
The solution set is
.
{ }
Your Turn:
Solve the equation. Check your answer.
–2.3 + m = 7
–2.3 + m = 7
+2.3
+ 2.3
m = 9.3
Check
–2.3 + m = 7
–2.3 + 9.3 7
7 7 
Since –2.3 is added to m, add 2.3
to both sides.
The solution set is {9.3}.
To check your solution, substitute 9.3
for m in the original equation.
Your Turn:
Solve the equation. Check your answer.
5
+z=
4
Since
is added to z add
to both sides.
The solution set is {2}.
Check
To check your solution,
substitute 2 for z in the
original equation.

Your Turn:
Solve the equation. Check your answer.
The reciprocal of
is
w is multiplied by
both sides by
w = 612
. Since
multiply
.
The solution set is {612}.
Check
To check your solution,
substitute 612 for w in the
original equation.
102
102

Example: Application
1
4
Ciro deposits of the money he earns from
mowing lawns into a college education fund. This
year Ciro added $285 to his college education
fund. Write and solve an equation to find out
how much money Ciro earned mowing lawns this
year.
Example: Continued
1
4
times
earnings
is
$285

e
=
$285
Write an equation to represent the
relationship.
4  1 e = 4  285
1 4
1
e = $1140
1
4
The reciprocal of 4 is 1 . Since e
is multiplied by 1 ,
4
multiply both sides by 4 .
1
The original earnings were $1140 .
Your Turn:
The distance in miles from the airport that a
plane should begin descending divided by 3
equals the plane’s height above the ground in
thousands of feet. A plane is 10,000 feet above the
ground. Write and solve an equation to find the
distance from the airport at which this plane
should begin descending.
Check It Out! Example 4 Continued
distance
d
divided by
3
is
height
÷
3
=
h
Write an equation to represent the
relationship.
3 d = 3 10
1  3
1 
d = 30
Substitute 10 for h. The reciprocal of 13
is 3 . Since d is multiplied by 13
1
multiply both sides by 3 .
1
At 10,000 feet altitude the decent should start 30,000 feet from
the airport.
Joke Time
• How was the Roman Empire cut in half?
• With a pair of Caesars.
• What kind of lighting did Noah use for the ark?
• Floodlights.
• Who invented fractions?
• Henry the 1/8th.
Assignment
• 2.1 Exercises Pg. 91 – 93: #10 – 68 even