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Solving Equations: Multiple-step
solutions
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Number Sense and Algebra
Solving Equations:
Multiple-step solutions
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PRESENTED BY ALGESTAR
Mathematics, Grade 9
[Title]
Solving Equations: Multiple-step solutions
Introduction
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Solving Equations: Multiple-step
solutions
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[Title]
Solving Equations: Multiple-step solutions
What you will learn
At the end of this lesson, you
will be able to
 solve equations that require multiple
steps to reach a final solution.
Equations can have multiple terms,
brackets, or fractions in them.
[Title]
Solving Equations: Multiple-step solutions
Agenda
• Background and importance
• Definitions and terms
• Back to basics
• Multiple-step solutions:
 Multiple terms
 Brackets
 Fractions
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Agenda
• Background and importance
• Definitions and terms
• Back to basics
• Multiple-step solutions:
 Multiple terms
 Brackets
 Fractions
[Title]
Solving Equations: Multiple-step solutions
ALGEBRA
• Branch of mathematics
that studies structure,
relation and quantity
• Uses letters or symbols to
represent numbers and
mathematical relations
Figure 1:
René Descartes.
• “Solving” algebraic
equations means
determining the actual
values the letters
represent
Portrait by Frans Hals,
1648.
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Solving Equations: Multiple-step
solutions
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[Title]
Solving Equations: Multiple-step solutions
Importance
Elementary algebra leads to




Number theory
Geometry
Trigonometry
Abstract algebra
Algebra is used in the fields of

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Architecture
Engineering
Graphic arts
Business/entrepreneurship
[Title]
Solving Equations: Multiple-step solutions
Agenda
• Background and importance
• Definitions and terms
• Back to basics
• Multiple-step solutions:
 Multiple terms
 Brackets
 Fractions
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Definitions
Variable Term
Term that contains a letter (variable)
Example:
7x, –4y, z
Constant Term
Term that contains ONLY a number
Example:
9, –2
[Title]
Solving Equations: Multiple-step solutions
Definitions
Coefficient
The factor by which a variable is
multiplied
Example:
–7 x
coefficient
variable
variable term
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Definitions
BEDMAS
A mnemonic (memory helper) used to
help with remembering the correct order
of operations
Brackets, Exponents, Division,
Multiplication, Addition, Subtraction
[Title]
Solving Equations: Multiple-step solutions
Definitions
Distributive Property
States that when a sum is multiplied by a
number, each value in the sum is
multiplied separately, and then the
products are added.
Example:
3(p – 4)
= 3(p +(–4))
= (3 x p) + (3 x –4)
= 3p – 12
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Solving Equations: Multiple-step
solutions
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[Title]
Solving Equations: Multiple-step solutions
Terms
“Simplify” means to add/subtract
the coefficients of like terms in an
expression.
e.g., 5x + 3x = 8x
“Solve” means to find the value of
the variable that makes the
equation true.
e.g., if x – 3 = 4, what is x?
[Title]
Solving Equations: Multiple-step solutions
Terms
“Inverse Operations” are
mathematical processes that are
the opposite of each other.
e.g., Addition is the inverse operation of
subtraction.
e.g., Division is the inverse operation of
multiplication.
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Solving Equations: Multiple-step
solutions
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[Title]
Solving Equations: Multiple-step solutions
Variable term
The variable term in the expression
–8x + 3 is
a) 3
b) –8x
[Title]
Solving Equations: Multiple-step solutions
Variable term
The variable term in the expression
–8x + 3 is
a) 3
b) –8x
This term contains the letter x (variable).
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Agenda
• Background and importance
• Definitions and terms
• Back to basics
• Multiple-step solutions:
 Multiple terms
 Brackets
 Fractions
[Title]
Solving Equations: Multiple-step solutions
Back to basics
Solving simple equations:
• First identify the operation acting on the
variable.
• Then isolate the variable by “undoing”
the operation (using the inverse
operation).
NOTE: It will look like terms are “moving”,
or being regrouped within the
equation.
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Back to basics
Solving simple equations:
Example:
x – 2 = 14
 2 is being subtracted from x
 Inverse of subtracting is adding on the other
side of the equation
x – 2 + 2 = 14 + 2
x = 16
(the 2 was regrouped with the 14)
[Title]
Solving Equations: Multiple-step solutions
Back to basics
Solving simple equations:
Example:
x is divided by –3
Inverse of dividing is multiplying on the other
side of the equation.
x = 8 x –3
x = –24
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Back to basics
Solve:
–2w = 42
a) w = 21
b) w = 44
c) w = –21
d) w = 40
[Title]
Solving Equations: Multiple-step solutions
Back to basics
Solve:
–2w = 42
a) w = 21
b) w = 44
c) w = –21
d) w = 40
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Back to basics
Solve: –2w = 42
w is multiplied by –2
Inverse of multiplying is dividing on the
other side
[Title]
Solving Equations: Multiple-step solutions
Agenda
• Background and importance
• Definition and terms
• Back to basics
• Multiple-step solutions:
 Multiple terms
 Brackets
 Fractions
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Multiple terms
Solve: –4x + 6 = 18
 Equation involves more than one
operation
 We need to undo EACH operation being
applied to the variable in the reverse
order to BEDMAS, until the variable is
isolated
[Title]
Solving Equations: Multiple-step solutions
Multiple terms
Look for terms that are added or
subtracted first (BEDMAS)
–4x + 6 = 18
The inverse of adding is subtracting. So,
move (regroup) the 6 to the right side of
the equation by subtracting it from 18.
–4x + 6 – 6 = 18 – 6
–4x = 12
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Solving Equations: Multiple-step
solutions
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[Title]
Solving Equations: Multiple-step solutions
Multiple terms
x is still being multiplied by –4
–4x = 12
Move the –4 by dividing on the
other side (BEDMAS)
[Title]
Solving Equations: Multiple-step solutions
Multiple terms
Solve:
3 + 3x – 10 = 5x + 31
 Decide which terms (i.e., variables
on one side, constants on the other)
will go on which side of the equals
sign
 Easier to go left to right through the
question, moving terms (regrouping
like terms together) as necessary.
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Solving Equations: Multiple-step
solutions
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[Title]
Solving Equations: Multiple-step solutions
Multiple terms
Solve: If we choose to have
constants on the left side:
3 + 3x – 10 = 5x + 31
(these terms don’t move)
3 + 3x – 10 = 5x + 31
(these terms need to be regrouped)
[Title]
Solving Equations: Multiple-step solutions
Multiple terms
Solve: 3 + 3x – 10 = 5x + 31
 Use inverse operations to regroup
3 – 10 – 31 = 5x – 3x
 Simplify in each group
–38 = 2x
 Divide by the coefficient of x
–19 = x
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Solving Equations: Multiple-step
solutions
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[Title]
Solving Equations: Multiple-step solutions
Multiple terms
Solve: 4x – 7 + 9x = –12 – 2x
Is this a correct next step?
4x + 9x – 2x = –12 + 7
a) Yes
b) No
[Title]
Solving Equations: Multiple-step solutions
Multiple terms
Solve: 4x – 7 + 9x = –12 – 2x
Is this a correct next step?
4x + 9x – 2x = –12 + 7
a) Yes
b) No
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Multiple terms
Solve: 4x – 7 + 9x = –12 – 2x
When you regroup –2x, which is on the
right side of the equation, it becomes +2x
on the left side.
4x + 9x + 2x = –12 + 7
15x = –5
[Title]
Solving Equations: Multiple-step solutions
Brackets
Solve: –3(5 – 6a) = 39
Simplify the equation first by removing
the brackets using the Distributive
Property
(–3 x 5) + (–3 x –6a) = 39
–15 + 18a = 39
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Brackets
 Now regroup terms by using inverse
operations in reverse
–15 – 39 = –18a
 Simplify
–54 = –18a
 Divide by the coefficient of ‘a’
3=a
[Title]
Solving Equations: Multiple-step solutions
Brackets
Given –15 + 18a = 39
Can I regroup as: 18a = 39 + 15?
a) Yes
b) No
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Brackets
Given –15 + 18a = 39
Can I regroup as: 18a = 39 + 15?
a) Yes
b) No
[Title]
Solving Equations: Multiple-step solutions
Brackets
Can I regroup as: 18a = 39 + 15?
 When regrouping terms, the variable
term (a) can be on the left or the right
side of the equation.
 One guideline for deciding is to let the
first term stay where it is, and move the
other terms accordingly.
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Brackets
• In this case, keeping the variable term
on the left means needing to move only
one term instead of two.
• The final solution will still be the same in
both cases, although when solving the
equation, the signs for the terms in each
step will be opposite.
18a = 54
a=3
[Title]
Solving Equations: Multiple-step solutions
Brackets
Solve: 3(2h – 5) + 4(3h + 2) = 11
 Distribute
6h – 15 + 12h + 8 = 11
 Regroup
6h + 12h = 11 + 15 – 8
 Simplify
18h = 18
 Divide by coefficient of variable (h)
h=1
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Fractions
Solve:
Contains a single fraction
 Regroup by moving +6 first
[Title]
Solving Equations: Multiple-step solutions
Fractions
 Move 2 from the denominator by
multiplying it on the other side
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Fractions
Solve:
More than one fraction
 Clear out the fractions first
 Make sure each term has a denominator
 Choose a common denominator
[Title]
Solving Equations: Multiple-step solutions
Fractions
 Common denominator = 10
 Multiply each term by the common
denominator and reduce the fractions
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Solving Equations: Multiple-step
solutions
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[Title]
Solving Equations: Multiple-step solutions
Fractions
Continue solving as a multiple-term
equation
Solve:
4x + 20 = 5
 Regroup 20 with 5 by subtracting
4x = 5 – 20
4x = –15
 Divide by coefficient of x
[Title]
Solving Equations: Multiple-step solutions
Fractions
The solution of
is 105.
a) True
b) False
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Fractions
The solution of
is 105.
a) True
b) False
[Title]
Solving Equations: Multiple-step solutions
Fractions
Solve
 Regroup 4 with 19 by subtracting
 Regroup 7 with 15 by multiplying
-q = 15 x 7
-q = 105
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Solving Equations: Multiple-step
solutions
Listen & Learn
[Title]
Solving Equations: Multiple-step solutions
Fractions
In the final solution, the variable
cannot have a negative coefficient.
 Divide by –1 (coefficient of q)
q = –105
[Title]
Solving Equations: Multiple-step solutions
Resources
Purplemath
 www.purplemath.com/modules/solvelin
.htm
Algebra.help
 www.algebrahelp.com/lessons
Math.com homeworkhelp
 www.math.com/school/subject2/lessons
/S2U3L1GL.html
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