Download Notes for Writing Equations

Solving Equations
 A-REI. 1
EXPLAIN each step in SOLVING a simple
equation as following from the equality of
numbers asserted at the previous step,
starting from the assumption that the
original equation has a
solution. CONSTRUCT a viable argument
to justify a solution method.
How can I write verbal expressions and equations
from algebraic forms?
2. How can I write algebraic expressions and
equations from verbal forms?
3. How are equations created to describe the
relationship between numbers and variables?
4. How do you solve linear equations and inequalities
in one-variable
1.
General little statements that
DO NOT have an equals sign!
 2x – 4
 2m³ + 5m
 Seven less than half a number
squared
 Twice the difference of a number
and five
Any statement that DOES have
an equals sign (or equivalent):
 4n + 7 = 11
½ b – 2 = 3
 Twice a number increased by four is
ten
 The difference of a number and two is
five
It means you will
take an expression
like 3x – 5, and
write it as: “the
difference of three
times a number and
five” or maybe “
five less than three
times a number”
If you are given the verbal sentence, you will
write it as an algebra equation or expression:
“half of a number increased by three is twelve”
would be written as : ½ x + 3 = 12
 Addition:
add,
plus, increased,
sum, more than
 Subtraction:
minus, less than,
decreased,
subtract, declines
•Multiplication:
multiply, per, per
person, each, each
customer,
installments,
percent, product,
of
•Division: divided
by, split, quotient
How would you write 4n + 3 as a verbal
expression?
(THINK-PAIR-SHARE)
Can you write the following in algebraic form?
“3 less than a number squared is the same as five”
1. Twice a number less than seven
2. Seven decreased by twice a number
3. Seven less than twice a number
4. Twice a number decreased by seven
 Two
more than 3 times
a number squared is
fourteen
 Three
m’s squared
plus two equals
fourteen
 Three
times a number
squared increased by two
is the same as fourteen
 It
cost $5 per hour to rent golf carts from
E-Z-GO. You need to rent golf carts for 10
hours. Write an equation. How much will
it cost you?
Pair Share!!!!
 What
are the variables involved in this
relationship?
 Which
variable is dependent?
 Which variable is independent?
 Choose
letters to represent your
variables
 What
variable is the question asking you
to find?
 (what are you looking for?)
 **HINT**
• This is the dependent variable and is always the
first letter you write; then you write the = sign.
 Make
a rule!
 (Say in words how you can find the
quantity for the variable you just wrote in
step 4)
 ASK YOURSELF: What calculations do I
need to use? (+, -, x, ÷)
 Addition:
add,
plus, increased,
sum, more than, in
addition
 Subtraction:
minus, less than,
decreased,
subtract, declines
•Multiply:
multiply, per,
each, installments,
percent, of, times
•Divide: divided
by, split, quotient,
per
 Write
an equation using your rule
 Test
it!
 ASK YOURSELF: Does your equation
make sense?






What are the variables? What are the variables?
Choose the letters. Choose the letters.
What are you looking for? What are you looking
for?
Make a rule. Make a rule.
Write an equation. Write an equation.
Does it make sense?
 The
number of hot dogs needed for the
picnic is two for each student
 How many hot dogs do we need for 150
students?
 The
amount of material needed to make
the curtains is 4 square yards per
window.
 How much material do we need for 5
windows?
 You
bring 30 cupcakes for your class
and there are s amount of students.
Write an equation to figure out how
many cupcakes each student (s) will
get.
 If there are 15 students in the class, how
many cupcakes will each student get?
 It
cost a flat fee of $25 plus $5 per hour to
rent golf carts from E-Z-GO. You need to
rent golf carts for 10 hours. Write an
equation. How much will it cost you?
9-12-’12
A. A number that changes constantly.
B. A letter or symbol that stands for a number.
C. A number that varies as you move away from
zero.
If I took one block away from
the left side, what would I
have to do to keep the scale
If I added 6 more apples to
the right side, how could I
keep the scale from tipping?
for Solving Addition and
Subtraction Equations
1. Look to see
what side the
variable is on.
2. Decide what
operation is
being shown.
3. Do the opposite
of that operation
to both sides.
4. Write what
number the
variable stands for
(ex: a = 2).
5. Check the solution by
“plugging” the number back
into the equation to see if the
sides are equal.
m + 18 = 34
-18 -18
m = 16
Look for variable
Do the opposite
Write what m is
m + 18 = 34
16 + 18 = 34 Plug in m
34 = 34
Solution is correct!
68 = y + 43
-43
-43
Look for variable
Do the opposite
Write what y is
25 = y
68 = y + 43
68 = 25 + 43 Plug in y
68 = 68
Solution is
correct!
t - 14 = 22
+ 14 +14
t = 36
t - 14 = 22
36 - 14 = 22
22 = 22
correct!
Look for variable
Do the opposite
Write what t is
Plug in t
Solution is
1.
2.
3.
4.
r + 5 = 18
m - 26 = 59
102 = x - 15
a + 39 = 56
r = 13
m = 85
117 = x
a = 17
To get the variable by itself,
which number needs to be
moved?
-5
To move the -5, you have to do
the opposite operation. What
operation will we use?
division
1.
2.
3.
4.
Draw “the river” to
separate the equation into
2 sides
Divide both sides by -5
Simplify
Check your answer
-5 -5
t = -12
-5(-12) = 60
1.
2.
3.
4.
Draw “the river”
Divide both sides by 6
Simplify
Check your answer
6 6
2.5 = n
15 = 6(2.5)
1.
2.
3.
4.
v = -126
v = -43
v = 43
v = 126
Answer Now
x
 12
4
You don’t like fractions? Let’s get rid
of them! 
“Clear the fraction”
by multiplying both sides of the
equation by the denominator.
1.
2.
3.
4.
Draw “the river”
Clear the fraction – multiply
both sides by 4
Simplify
Check your answer
x
 12
4
x
4 ·  12 ·
4
4
x
=
-48
48
 12
4
1.
2.
3.
4.
5.
6.
Draw “the river”
Clear the fraction – multiply
both sides by 3
Simplify
Divide both sides by 2
Simplify
Check your answer
2x
3
2x
3·
= 18 · 3
3
2(27)
 18
3
2x = 54
2
2
x = 27
3b
 12
5
1.
2.
3.
4.
Multiply by 3
Multiply by 5
Multiply by -12
Multiply by -5
Answer Now
4b
 8
7
1.
2.
3.
4.
b = -56
b = -14
b = 14
b = 56
Answer Now
Using a Verbal Model
JET PILOT A jet pilot is flying from Los Angeles, CA to Chicago,
IL at a speed of 500 miles per hour. When the plane is 600
miles from Chicago, an air traffic controller tells the pilot that
it will be 2 hours before the plane can get clearance to land.
The pilot knows the speed of the jet must be greater then 322
miles per hour or the jet could stall.
a. At what speed would the jet have to fly to arrive in Chicago in
2 hours?
b. Is it reasonable for the pilot
to fly directly to Chicago at
the reduced speed from part
(a) or must the pilot take
some other action?
Using a Verbal Model
a. At what speed would the jet have to fly to arrive in Chicago in 2 hou
You can use the formula (rate)(time) = (distance) to write a verbal model.
SOLUTION
VERBAL
MODEL
Speed of •
jet
LABELS
Speed of jet =x
(miles per hour)
Time = 2
(hours)
Time
=
Distance to
travel
(miles)
Distance to travel 600
=
ALGEBRAIC
MODEL
2
x = 600
x =
30
0
To arrive in 2 hours, the pilot would have to slow the jet down to 300 miles per hour.
Using a Verbal Model
b. Is it reasonable for the pilot to fly directly to Chicago
at 300 miles per hour or must the pilot take some
other action?
It is not reasonable for the
pilot to fly at 300 miles per
hour, because the jet could
stall. The pilot should take
some other action, such as
circling in a holding pattern, to
use some of the time.
Writing an Algebraic Model
You and three friends are having a dim sum lunch at a
Chinese restaurant that charges $2 per plate. You
order lots of plates. The waiter gives you a bill for
$25.20, which includes tax of $1.20. Use mental math
to solve the equation for how many plates your group
ordered.
SOLUTION
Understand the problem
situation before you begin. For
example, notice that tax is
added after the total cost of
the dim sum plates is figured.
Writing an Algebraic Model
VERBAL
MODEL
Cost per •
plate
LABELS
Cost per plate =2
Number of
plates
=
Number of plates p=
Bill
–
(dollars)
(plates)
Amount of bill = 25.20 (dollars)
Tax
=
ALGEBRAIC
MODEL
1.20
(dollars)
2 p = 25.20– 1.20
2p =
p =
24.00
12
Your group ordered 12 plates of food costing $24.00.
Tax