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Solving Equations
by Lauren McCluskey
Credits
Prentice Hall Algebra I
Solving One-Step Equations:
“An
equation is like a balance scale
because it shows that two
quantities are equal. The scales
remained balanced when the same
weight is added (or removed from)
to each side.”
x+2=5
x + (-5) = -2
What does it mean to Solve an
equation?
“To
solve an equation containing a
variable, you find the value (or values)
of the variable that make the equation
true.”
“Get the variable alone on one side of
the equal sign…using inverse
operations, which are operations that
undo each other.”
Inverse Operations
Addition and Subtraction are inverse
operations because they undo each
other.
Multiplication and division are inverse
operations because they undo each
other.
Properties of Equality:
• Addition Property of Equality: “For
every real number a, b, and c, if a = b,
then a + c = b + c.”
• Subtraction Property of Equality: “For
every real number a, b, and c, if a = b,
then a - c = b - c.”
Properties of Equality:
• Multiplication Property of Equality:
“For every real number a, b, and c, if a
= b, then a*c = b*c.”
• Division Property of Equality: “For
every real number a, b, and c, if a = b,
then a/c = b/c.”
Using Reciprocals:
2/3x = 12
In order to solve the equation above,
you need to divide by 2/3.
Remember: To divide a fraction,
you multiply by its reciprocal. In
other words: flip it!
3/2*2/3x= 1x
3/2* 12/1= 18
So x = 18.
Examples:
1) a - 4 = -18
-14
2) b + 24 = 19
-5
3) v/ 3 = -4
-12
4) 15c = 90
6
5) -1/5 r = -4
+20
Try It!
1) d + (-4) = -7
2) -54 = q - 9
3) m / -4 = 13
4) -75 = -15x
5) 2/3 n = 14
Check your answers:
1) -3
2) -45
3) -52
4) +7
5) +21
Solving Two-Step Equations
“A two-step equation is an equation
that involves two operations.”
PEMDAS tells us to multiply or divide
before we add or subtract, but to solve
equations, we do just the opposite: we
add or subtract before we multiply
or divide.
Try It!
1) 7 = 2y - 3
2) 6a + 2 = -8
3) x/9 - 15 = 12
4) -x + 7 = 12
5) a -5 = -8
6) 4 = -c + 11
Check your answers:
1) 7 = 2y - 3
+3
+3
10 = 2y
2
2
5= y
2) 6a + 2 = - 8
-2
-2
6a = -10
6
6
a = -1 2/3
To Solve Multi-Step Equations:
1) “Clear the equation of
fractions and decimals.”
2) Apply the Distributive
Property as needed.
3) “Combine like terms.”
4) “Undo addition and
subtraction.”
5) “Undo multiplication and
division.”
Example:
1/
5
+ 3w/ 15 = 4/5
To clear fractions: * every term by 5
5 * 1/5 = 1; 5* 3w/15 = 9w; 5 * 4/5= 4
So 1 + 9w = 4
Undo addition:
-1
-1
9w = 3
Undo multiplication:
9w / 9 = 3 / 9
So w = 1/3
Try It!
a
1) /7
9y
2) /
-
5/
7
6/
7
=
3
9
14 + /7 = /14
2
3
71
3) /3 + /k = /12
Check your answers:
1) a = 11
2) y = 1/3
3) k = 4/7
Example:
0.11p + 1.5 = 2.49
Clear decimals by * by 100
100 * 0.11= 11; 100 * 1.5 = 150; 100 * 2.49 = 249
So 11p + 150 = 249
Undo addition:
-150
-150
11p = 99
Undo multiplication:
11p/11 = 99/11
p=9
Try it!
1) 25.24 = 5y + 3.89
2) 0.25m + 0.1m = 9.8
3) 26.54 - p = 0.5(50 - p)
Check your answers:
1) y = 4.27
2) m = 28
3) p = 3.08
Try it!
1) -4(x + 6) = -40
2) m + 5(m -1) = 7
3) 1/4(m - 16 ) = 7
Check your answers:
1) x = 4
2) m = 2
3) m = 12
Equations with Variables on Both
Sides:
Use the Addition or Subtraction
property of Equality to get the
variables on one side of the
equation.
Example:
4p - 10 = p + 3p -2p
Combine like terms:
p + 3p - 2p = 2p
Use the subtraction property
of equality:
4p - 10 = 2p
-2p
-2p
2p -10 = 0
Example: cont.
2p - 10 = 0
Undo subtraction:
+10 +10
2p = 10
Undo multiplication:
2p / 2 = 10/ 2
p=5
Try It!
1)6b + 14 = -7 - b
2) -36 + 2w = -8w + w
3) 30 - 7z = 10z - 4
Check your answers:
1) b = -3
2) w = 4
3) z = 2
Identity or No Solution:
“An
equation has no solution if no
value of the variable makes the
equation true.”
“An equation that is true for every
value of the variable is an identity.”
2.5: Defining One Variable in
Terms of Another:
 “Some problems involve two or
more unknown quantities. To
solve such problems, first
decide which unknown quantity
the variable will represent. Then
express the other unknown
quantity in terms of that
variable.”
Example:
“The width of a rectangle is 2 cm
less than its length. The perimeter
of the rectangle is 16cm. What is
the length of the rectangle?”
Let l = length
Let l - 2 = width
P= 2l + 2w
So 2(l) + 2(l -2) = 16cm
Example: cont.
2l + 2(l-2) = 16 cm
2l + 2l - 4 = 16cm
4l - 4 = 16 cm
+4 +4
4l = 20 cm
4l / 4 = 20/ 4 so l = 5
Try it!
The length of a rectangle
is 6 more than 3 times as
long as the width. The
perimeter is 36 m. What
are the measurements?
Check your answer:
l = 15m
w = 3m
OR: 15m x 3m
Consecutive Integers:
“Consecutive integers differ by 1.”
Consecutive even or consecutive
odd integers differ by 2.
Example:
The sum or two consecutive odd
integers is 84. What are the
integers?
Consecutive Integers
Let x = the 1st integer
Let x + 2 = the 2nd integer
x + x + 2 = 84
2x + 2 = 84
-2 -2
2x = 82
x = 41; x + 2 = 43
So the integers are 41 and 43.
Try It!
The sum of three consecutive
integers is 48. What are the
integers?
Check your answer:
The integers are:
15, 16, and 17.
Rate* Times = Distance
 When the distances covered
are equal, we can set the two
expressions equal to each
other and solve for x.
 When the distances combine
to make up the total distance,
we can add the expressions,
set it equal to the total
distance, and solve for x.
Try It!
Adapted from Prentice Hall:
1) A group of campers left the
campsite in a canoe going
10km/h. Two hours later,
another group left in a motor
boat going 22km/h. How long
did it take the second group to
catch up?
Try It!
2) On his way to work, your
uncle averaged 20 mph. On his
way home, he averaged 40mph.
If the total time was 1 1/2hours,
how long did it take him to
drive to work?
Try It!
3) Sarah and John left Perryville
going in opposite directions.
Sarah drives 12mph faster than
John. After 2 hours, they are
176 miles apart. Find Sarah’s
and John’s speeds.
Check your answers:
1)1 3/4 hours
2) 1 hour
3) John= 33mph
Sarah= 45mph