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New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative
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Solving Equations
8th Grade
2012-12-17
www.njctl.org
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Table of Contents
Inverse Operations
One Step Equations
Two Step Equations
Multi-Step Equations
More Equations
Transforming Formulas
Click on a topic to go to
that section.
Inverse Operations
Return to
Table of
Contents
What is an equation?
An equation is a mathematical statement, in symbols,
that two things are exactly the same (or equivalent).
Equations are written with an equal sign, as in
2+3=5
9–2=7
Equations can also be used to state the equality of two
expressions containing one or more variables.
In real numbers we can say, for example, that for any given
value of x it is true that
4x + 1 = 14 - 1
If x = 3, then
4(3) + 1 = 14 - 1
12 + 1 = 13
13 = 13
When defining your variables, remember...
Letters from the beginning of the alphabet like a, b, c... often
denote constants in the context of the discussion at hand.
While letters from end of the alphabet, like x, y, z..., are
usually reserved for the variables, a convention initiated by
Descartes.
Try It!
Write an equation with a variable and have a classmate
identify the variable and its value.
An equation can be
compared to a balanced
scale.
Both sides need to
contain the same
quantity in order for it to
be "balanced".
For example, 20 + 30 = 50 represents an equation because
both sides simplify to 50.
20 + 30 = 50
50 = 50
Any of the numerical values in the equation can be
represented by a variable.
Examples:
20 + c = 50
x + 30 = 50
20 + 30 = y
Why are we Solving Equations?
First we evaluated expressions where we were given the value
of the variable and had to find what the expression simplified
to.
Now, we are told what it simplifies to and we need to find the
value of the variable.
When solving equations, the goal is to isolate the variable on
one side of the equation in order to determine its value (the
value that makes the equation true).
In order to solve an equation containing a variable, you
need to use inverse (opposite/undoing) operations on
both sides of the equation.
Let's review the inverses of each operation:
Addition
Multiplication
Subtraction
Division
There are four properties of equality that we will use to solve
equations. They are as follows:
Addition Property
If a=b, then a+c=b+c for all real numbers a, b, and c. The same
number can be added to each side of the equation without
changing the solution of the equation.
Subtraction Property
If a=b, then a-c=b-c for all real numbers a, b, and c. The same
number can be subtracted from each side of the equation
without changing the solution of the equation.
Multiplication Property
If a=b, and c=0, then ac=bc for all real numbers ab, b, and c.
Each side of an equation can be multiplied by the same nonzero
number without changing the solution of the equation.
Division Property
If a=b, and c=0, then a/c=b/c for all real numbers ab, b, and c.
Each side of an equation can be divided by the same nonzero
number without changing the solution of the equation.
To solve for "x" in the following equation...
x + 7 = 32
Determine what operation is being shown (in this case, it is
addition). Do the inverse to both sides.
x + 7 = 32
-7
-7
x = 25
In the original equation, replace x with 25 and see if it makes
the equation true.
x + 7 = 32
25 + 7 = 32
32 = 32
For each equation, write the inverse operation needed to solve for
the variable.
a.) y + 7 = 14 subtract
move7
c.) 5s = 25
mov
divide
by 5
move
e
b.) a - 21 = 10
d.)
x =5
12
add
21
move
move
multiply
by 12
Think about this...
To solve c - 3 = 12
Which method is better? Why?
Kendra
Ted
Added 3 to each side of
the equation
Subtracted 12 from each side,
then added 15.
c - 3 = 12
+3 +3
c = 15
c - 3 = 12
-12 -12
c - 15 = 0
+15 +15
c = 15
Think about this...
In the expression
To which does the "-" belong?
Does it belong to the x? The 5? Both?
The answer is that there is one negative so it is used once with
either the variable or the 5. Generally, we assign it to the 5 to
avoid creating a negative
Touchvariable.
to reveal answer
So:
1
What is the inverse operation needed to solve
this equation?
7x = 49
A
Addition
B
Subtraction
C
Multiplication
D
Division
2
What is the inverse operation needed to solve
this equation?
x - 3 = -12
A
Addition
B
Subtraction
C
Multiplication
D
Division
One Step Equations
Return to
Table of
Contents
To solve equations, you must work backwards through the
order of operations to find the value of the variable.
Remember to use inverse operations in order to isolate the
variable on one side of the equation.
Whatever you do to one side of an equation, you MUST do to
the other side!
Examples:
y + 9 = 16
- 9 -9
y=7
6m = 72
6
6
m = 12
The inverse of adding 9 is subtracting 9
The inverse of multiplying by 6 is dividing by 6
Remember - whatever you do to one side of an equation, you
MUST do to the other!!!
One Step Equations
Solve each equation then click the box to see work & solution.
x - 8 = -2
+8 +8
click to
show
x=
6
inverse operation
x + 2 = -14
-2 -2
click to show
x = -16
inverse operation
x+5=3
-5 -5
click to show
x = -2
inverse operation
2=x-6
+6
+6
click to show
8=x
inverse operation
7=x+3
-3
-3
4click
= xto show
inverse operation
15 = x + 17
-17
-17
-2 click
= x to show
inverse operation
One Step Equations
3x = 15
3
3
click to show
x = operation
5
inverse
x
(2) 2 = 10 (2)
x = 20
click to show
inverse operation
-4x = -12
-4click to-4
show
inverse
x = 3operation
-25 = 5x
5
5
click to show
-5 = operation
x
inverse
(-6)
x = 36 (-6)
-6
x = -216
click to show
inverse operation
3
Solve.
x - 6 = -11
4
Solve.
j + 15 = -17
5
Solve.
-115 = -5x
6
Solve.
x
9
= 12
7
Solve.
51 = 17y
8
Solve.
w - 17 = 37
9
Solve.
x
-3 =
7
10
Solve.
23 + t = 11
11
Solve.
108 = 12r
Two-Step Equations
Return to
Table of
Contents
Sometimes it takes more than one step to solve an equation.
Remember that to solve equations, you must work backwards
through the order of operations to find the value of the
variable.
This means that you undo in the opposite order (PEMDAS):
1st: Addition & Subtraction
2nd: Multiplication & Division
3rd: Exponents
4th: Parentheses
Whatever you do to one side of an equation, you MUST do to
the other side!
Examples:
3x + 4 = 10
- 4 - 4 Undo addition first
3x = 6
3 3 Touch
Undotomultiplication
reveal answersecond
x=2
-4y - 11 = -23
+ 11 +11
Undo subtraction first
-4y = -12
-4
-4 Touch to
Undo
multiplication
second
reveal
answer
y=3
Remember - whatever you do to one side
of an equation, you MUST do to the other!!!
Two Step Equations
Solve each equation then click the box to see work & solution.
6-7x = 83
-6
-6
-7x = 77
-7 -7
x = -11
3x + 10 = 46
- 10 -10
3x = 36
3
3
x = 12
-2x + 3 = -1
- 3 -3
-2x = -4
-2 -2
x=2
9 + 2x = 23
-9
-9
2x = 14
2
2
x=7
-4x - 3 = 25
+3 +3
-4x = 28
-4 -4
x = -7
8 - 2x = -8
-8
-8
-2x = -16
-2
-2
x=8
Walter is a waiter at the Towne Diner. He earns a daily wage of
$50, plus tips that are equal to 15% of the total cost of the
dinners he serves. What was the total cost of the dinners he
served if he earned $170 on Tuesday?
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
12
Solve the equation.
5x - 6 = -56
13
Solve the equation.
16 = 3m - 8
14
Solve the equation.
15
Solve the equation.
5r - 2 = -12
16
Solve the equation.
12 = -2n - 4
17
Solve the equation.
18
Solve the equation.
19
What is the value of n in the equation
0.6(n + 10) = 3.6?
A
-0.4
B
5
C
-4
D
4
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011
20
In the equation
n is equal to?
A
8
B
2
C
D
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
21
Which value of x is the solution of the equation
?
A
B
C
D
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June,
2011
22
Two angles are complementary. One angle has
a measure that is five times the measure of the
other angle. What is the measure, in degrees, of
the larger angle?
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Multi-Step Equations
Return to
Table of
Contents
Steps for Solving Multiple Step Equations
As equations become more complex, you should:
1. Simplify each side of the equation.
(Combining like terms and the distributive property)
2. Use inverse operations to solve the equation.
Remember, whatever you do to one side of an equation,
you MUST do to the other side!
Examples:
-15 = -2x - 9 + 4x
-15 = 2x - 9 Combine Like Terms
+9
+9
Undo Subtraction first
-6 = 2x
Touch to reveal answer
2 2
Undo Multiplication second
-3 = x
7x - 3x - 8 = 24
4x - 8 = 24 Combine Like Terms
+ 8 +8 Undo Subtraction first
Touch to reveal answer
4x = 32
4
4
Undo Multiplication second
x=8
Now try an example. Each term is infinitely cloned
so you can pull them down as you solve.
-7x
+ 3 + 6x = -6
ans
wer
Now try another example. Each term is infinitely cloned
so you can pull them down as you solve.
6x - 5 + x = 44
ans
wer
Always check to see that both sides of the equation are
simplified before you begin solving the equation.
Sometimes, you need to use the distributive property in
order to simplify part of the equation.
Distributive Property
For all real numbers a, b, c
a(b + c) = ab + ac
a(b - c) = ab - ac
Examples
5(20 + 6) = 5(20) + 5(6)
9(30 - 2) = 9(30) - 9(2)
3(5 + 2x) = 3(5) + 3(2x)
-2(4x - 7) = -2(4x) - (-2)(7)
Examples:
5(1 + 6x) = 185
5 + 30x = 185
Distribute the 5 on the left side
-5
-5
Undo addition first
30x
= 180
Move
to reveal answer
30
30
Undo multiplication second
x=6
2x + 6(x - 3) = 14
2x + 6x - 18 = 14 Distribute the 6 through (x - 3)
8x - 18 = 14
Combine Like Terms
+18 +18
Undo subtraction
to reveal answer
8x = Move
32
8
8
Undo multiplication
x=4
Now show the distributing and solve...(each number/ symbol
is infinitely cloned, so click on it and drag another one down)
5 ( -2 + 7x ) = 95
Now show the distributing and solve...(each number/ symbol
is infinitely cloned, so click on it and drag another one down)
6 ( -2x + 9 ) = 102
23
Solve.
3 + 2t + 4t = -63
24
Solve.
19 = 1 + 4 - x
25
Solve.
8x - 4 - 2x - 11 = -27
26
Solve.
-4 = -27y + 7 - (-15y) + 13
27
Solve.
9 - 4y + 16 + 11y = 4
28
Solve.
6(-8 + 3b) = 78
29
Solve.
18 = -6(1 - 1k)
30
Solve.
2w + 8(w + 3) = 34
31
Solve.
4 = 4x - 2(x + 6)
32
Solve.
3r - r + 2(r + 4) = 24
33
What is the value of p in the equation
2(3p - 4) = 10?
A
1
B
2 1/3
C
3
D
1/3
From the New York State Education Department. Office of Assessment Policy, Development
and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra;
accessed 17, June, 2011.
More Equations
Return to
Table of
Contents
Remember...
1. Simplify each side of the equation.
2. Solve the equation.
(Undo addition and subtraction first, multiplication and division second)
Remember, whatever you do to one side of an equation, you
MUST do to the other side!
Examples:
3
5x= 6
5
3
3x= 6 5
Multiply both sides by the
5
3
reciprocal
Click to reveal steps
x = 30
3
x = 10
2x - 3 = -14 + x
5
-x
-x
Subtract x from both sides
x - 3 = -14
5 Click to reveal steps
+3
+3
Undo Subtraction
1
x= 5
There is more than one way to solve an equation with distribution.
3 (-3 + 3x) = 72
5
5
Multiply by the reciprocal
Multiply by the LCM
3
72
(-3 + 3x) =
5
5
5 3
72
(-3 + 3x) =
3 5
5
-3 + 3x = 24
+3
+3
3x = 27
3
3
x=9
5
3
5
3
(-3 + 3x) =
5
72
5
3
(-3 + 3x) =
5
72 5
5
3(-3 + 3x) = 72
-9 + 9x = 72
+9
+9
9x = 81
9
9
x=9
34
Solve
35
Solve
36
Solve
37
Solve
38
Solve
7(2x +9) = -3(21)
39
Solve
Transforming
Formulas
Return to
Table of
Contents
Formulas show relationships between two or more
variables.
You can transform a formula to describe one quantity
in terms of the others by following the same steps as
solving an equation.
Example:
Transform the formula d = r  t to find a formula
for time in terms of distance and rate.
What does "time in terms of distance and rate" mean?
d=r t
r
r
Divide both sides by r
dSlide to reveal steps
=t
r
Examples
V = l wh
Solve for w
V =w
lh
P = 2l + 2w
-2w
-2w
P - 2w = 2l
2
2
P - 2w = l
2
Slide to reveal
steps
Solve for l
Slide to
reveal
steps
Example:
To convert Fahrenheit temperature to Celsius, you use the
formula:
5
C = 9 (F - 32)
Transform this formula to find Fahrenheit temperature in
terms of Celsius temperature. (see next page)
Solve the formula for F
C = 5 (F - 32)
9
C = 5 F - 160
9
9
160
+ 160 Slide
+ to
9 reveal 9
)= 5 F
9 (C + 160
steps
5
9
9
9 C + 32 = F
5
9
5
Transform the formula for area of a circle to find radius
when given Area.
A=
r
2
A = r2
Slide
to
A reveal
=r
answer
Solve the equation for the given variable.
m
p
=
for p
n
q
(q) m = p (q)
n
q
Move to
mqreveal
p
=
n steps
2(t + r) = 5 for t
2(t + r) = 5
2
2
Move5to
t + rreveal
=
2
steps
-r -r
t=
5
r
2
40
The formula I = prt gives the amount of simple
interest, I, earned by the principal, p, at an
annual interest rate, r, over t years.
Solve this formula for p.
A
p = Irt
B
p=
C
D
Ir
t
p= l
rt
p = It
r
41
A satellite's speed as it orbits the Earth is
2
found using the formula v = Gm .
r
In this formula, m stands for the mass of the
Earth. Transform this formula to find the mass
of the Earth.
A
B
2 - r
v
m=
G
2
m = rv - G
2
C
D
m= v-r
G
2
m = rv
G
42
Solve for t in terms of s
4(t - s) = 7
A
B
C
D
t= 7+s
4
t = 28 + s
t= 7 -s
4
t= 7+s
4
43
Solve for w
A = lw
A
w = Al
B
w= A
l
w= l
A
C
44
Solve for h
A
B
C
D
45
Which equation is equivalent to 3x + 4y = 15?
A
y = 15 − 3x
B
y = 3x − 15
C
y = 15 – 3x
4
y = 3x – 15
4
D
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011
.
46
, b ≠ 0, then x is equal to
If
A
B
C
D
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June,
2011