Transcript solving equations - MrsFaulkSaysMathMatters

Objective
The student will be able to:
solve equations using addition and
subtraction.
Designed by Skip Tyler, Varina High School
1) Solve r + 16 = -7
Think of this equation as a balance scale.
Whatever you do to one side has to be done to the
other to keep it balanced!
1) Solve r + 16 = -7
To solve, you must get the variable by
itself.
What number is on the same side as r?
16
To get r by itself, we must undo the
“add 16”. What is the opposite of
addition?
Subtract 16
1.
2.
3.
4.
1) Solve r + 16 = -7
Draw “the river” to
16
-16
separate the equation
into 2 sides
r
=
-23
Subtract 16 from both
sides
Simplify vertically
Check your answer
by substituting your
answer back into the
problem
-23 + 16 = -7
2) Solve x + 2 = -3
Get the variable by itself. What is your
first step?
1.
2.
3.
4.
Add 2 to both sides
Subtract 2 from both sides
Add 3 to both sides
Subtract 3 from both sides
Answer Now
1.
2.
3.
4.
2) Solve x + 2 = -3
Draw “the river” to
2
2
separate the equation
into 2 sides
x
=
-5
Subtract 2 from both
sides
Simplify vertically
Check your answer
by substituting your
answer back into the
problem
-5 + 2 = -3
On homework and tests, be sure to check your work!!
There is no reason why you should miss a problem!
3) Solve 8 = m - 3
1.
2.
3.
4.
m=5
m = 11
m = 24
m = 8/3
Answer Now
1.
2.
3.
4.
3) Solve 8 = m - 3
Draw “the river” to
+
3
+
3
separate the equation
into 2 sides
11=
m
Add 3 to both sides
Simplify vertically
Check your answer
by substituting your
answer back into the
problem
8 = 11 - 3
When solving equations, we want to
eliminate double signs.
y + (-3) = 8
is rewritten as
y–3=8
p – (-5) = 6
is rewritten as
p+5=6
As a general rule, replace “+ (- )” with “–” and “– (- )” with “+”.
This will make things less confusing in the future!
4) Solve y + (-3) = 7
1.
2.
3.
4.
5.
Draw “the river” to
separate the equation
into 2 sides
Eliminate the double
sign
Add 3 to both sides
Simplify vertically
Check your answer
by substituting your
answer back into the
problem
y–3 =7
+ 3 +3
y
= 10
10 + (-3) = 7
5) Solve. -x - (-2) = 1
1.
2.
3.
4.
5.
6.
Draw “the river” to
separate the equation into
2 sides
Eliminate the double sign
Subtract 2 from both sides
Simplify vertically
We haven’t gotten x by
itself. If we read this
aloud, it is “the opposite
of x equals -1”. What
would x be equal?
Check your answer
-x + 2 = 1
-2 -2
-x
= -1
x
= 1
-(1) + 2 = 1
Solve -y – (-3) = 7
1.
2.
3.
4.
y = 10
y=4
y = -10
y = -4
Answer Now
3-4: Solving Equations by
Multiplying or Dividing
Designed by Skip Tyler, Varina High School
1) Solve. -5t = 60
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.
1) Solve -5t = 60
-5 -5
Draw “the river” to
t = -12
separate the equation
into 2 sides
Divide both sides by -5
Simplify
Check your answer
-5(-12) = 60
1.
2.
3.
4.
2) Solve 15 = 6n
6
6
Draw “the river”
Divide both sides by 6
2.5 = n
Simplify
Check your answer
15 = 6(2.5)
3) Solve -3v = -129
1.
2.
3.
4.
v = -126
v = -43
v = 43
v = 126
Answer Now
x
 12
4) Solve
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.
x
4) Solve  12
4
Draw “the river”
x
Clear the fraction –
4 ·  12 · 4
multiply both sides by 4
4
Simplify
Check your answer
x = -48
48
 12
4
1.
2.
2x
5) Solve
= 18
3
2x
3·
= 18 · 3
Draw “the river”
3
Clear the fraction –
3.
4.
5.
6.
multiply both sides by 3
Simplify
Divide both sides by 2
Simplify
Check your answer
2x = 54
2
2
x = 27
2(27)
 18
3
6) Which step clears the fraction in
3b
 12
5
1.
2.
3.
4.
Multiply by 3
Multiply by 5
Multiply by -12
Multiply by -5
Answer Now
7) Solve
1.
2.
3.
4.
b = -56
b = -14
b = 14
b = 56
Answer Now
4b
 8
7
3-5: Solving Two-Step
Equations
What is a Two-Step Equation?
An equation written in the form
Ax + B = C
Examples of Two-Step Equations
a) 3x – 5 = 16
b) y/4 + 3 = 12
c) 5n + 4 = 6
d) n/2 – 6 = 4
Steps for Solving
Two-Step Equations
1.
Solve for any Addition or Subtraction on the
variable side of equation by “undoing” the
operation from both sides of the equation.
2.
Solve any Multiplication or Division from variable
side of equation by “undoing” the operation from
both sides of the equation.
Opposite Operations
Addition  Subtraction
Multiplication  Division
Helpful Hints?
Identify what operations are on the
variable side. (Add, Sub, Mult, Div)
“Undo” the operation by using
opposite operations.
Whatever you do to one side, you
must do to the other side to keep
equation balanced.
Ex. 1: Solve 4x – 5 = 11
4x – 5 =
+5
4x
=
4
x=5
15
+5 (Add 5 to both sides)
20 (Simplify)
4 (Divide both sides by 4)
(Simplify)
Try These Examples
1. 2x – 5 = 17
2. 3y + 7 = 25
3. 5n – 2 = 38
4. 12b + 4 = 28
Check your answers!!!
1. x = 11
2. y = 6
3. n = 8
4. b = 2
Ready to Move on?
Ex. 2: Solve x/3 + 4 = 9
x/3 + 4 = 9
-4 -4
x/3 = 5
(Subt. 4 from both sides)
(Simplify)
(x/3)  3 = 5  3 (Mult. by 3 on both sides)
x = 15
(Simplify)
Try these examples!
1. x/5 – 3 = 8
2. c/7 + 4 = 9
3. r/3 – 6 = 2
4. d/9 + 4 = 5
Check your answers!!!
1. x = 55
2. c = 35
3. r = 24
4. d = 9
Time to Review!
•
•
•
Make sure your equation is in the form Ax + B = C
Keep the equation balanced.
Use opposite operations to “undo”
•
Follow the rules:
1. Undo Addition or Subtaction
2. Undo Multiplication or Division
Created by: Jimmy Frost
3-6: Writing Two-Step
Equations
What you will learn:
Write verbal sentences as
two-step equations
Solve verbal problems by
writing and solving twostep equations
In Chapter 1, you learned how to write verbal phrases as
expressions.
Phrase: the sum of 4 times some number and 99
Expression:
4n
+
99
An equation is a statement that two expressions are equal. The
expressions are joined with an equals sign. Look for the words
equals or is equal to when you translate sentences into
equations
Sentence: The sum of 4 times some number and 99 is 299
Equation:
4n + 99
= 299
Translate sentences into equations:
Six more than twice a number is -20
2n + 6 = -20
Eighteen is 6 less than four times a number
18 = 4n - 6
The quotient of a number and 5, increased
by 8, is equal to 14
n + 8 = 14
5
Remember: Read carefully what the problem is saying.
Don’t mix up quotient () and product ().
Translate and Solve an Equation
Seven more than three times a number is 31. Find the number.
Words: Seven more than three times a number is 31
Variables: Let n = the number
Equation: 3n + 7 = 31
3n + 7 = 31
3n + 7 -7 = 31 - 7
3n = 24
3
3
n=8
Check:
3(8) + 7 = 31
24 + 7 = 31
31 = 31
Write and solve a two-step equation
Suppose you are saving money to buy a scooter that costs
$100. You have already saved $60 and plan to save $5 each
week. How many weeks will you need to save?
Explore: You have already saved $60 and you plan to save $5
a week until you have $100.
Plan: Organize the data for the first few weeks in a table.
Notice the pattern.
Write an equation
To represent the
Week
Amount
Situation. Let x
Equal the number
0
5(0)+60=60
of weeks.
1
2
3
5(1)+60=65
5(2)+60=70
5(3)+60=75
5x + 60 = 100
Now solve
5x + 60 = 100
5x + 60 -60 = 100 - 60
5x = 40
5
5
x=8
Check:
5(8) + 60 = 100
40 + 60 = 100
100 = 100
You need to save $5 each week for 8 weeks.
In the 2000 Summer Olympics, the United States won 9 more
medals than Russia. Together they won 185 medals. How many
more medals did the United States win?
Words: Together they won 185 medals
Variables: Let x = number of medals Russia won
Then x + 9 = number of medals won by U.S.
Equation: x + (x+9) = 185
Solve:
x + (x + 9) = 185
2x + 9 = 185
2x + 9 -9 = 185 -9
2x = 176
2
2
x = 88
Check:
88 + (88+9) = 185
88 + 97 = 185
185 = 185
Russia won 88 medals
U.S. won 97 medals