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2-5 Solving Equations with the
Variable on Each Side
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
Since variables represent numbers, you can add or
subtract variable terms. Remember the basic rule for
solving equations:
Whatever you do to one side of the equal sign, you
must also do to the other side of the equal sign.
To solve equations that have variables on both sides of
the equal sign, first collect the variable terms on one
side of the equation.
Collecting the variable terms on the side with the
greater variable coefficient will result in a positive
variable.
An equation is not solved if the variable is negative.
Solve 6x  22  3x  31.
Check
6x  22  3x  31
Write the problem. 6x  22  3x  31
 3x
 3x
Collect variable
61  22  31  31
terms on one side. 9x  22  31
6  22  3  31
Undo addition
 22  22

28
 28
and simplify.
9x  9
What is the
Undo multiplication
9 9
greater variable
x 1
and simplify.
coefficient?
On a test, do not check your answers until you have
completed all the problems. Then formally check your
equations. This is good test taking strategy so you do not
spend too much time on one problem and fail to finish the
test.
Solve.
Write problem.
Distribute and
combine like terms.
Collect variable
terms on one side.
Isolate the variable using
inverse operations.
Copy in your
spiral
notebook!
6x  3  3x  3x  2
6x  18  3x  3x  6
9x  18  3x  6
 3x
 3x
6x  18   6
 18  18
6x  24
6
6
x  4
Please take a highlighter or red pencil and draw a
line through the equal signs for the problem you
just copied. Does it produce a straight line?
Example 1
64  12w  6w
 12w  12w
64  18w
18 18
32
w
9
Solve.
Example 2
7 x  19  2x  55
 2x
 2x
9x  19  55
 19  19
9x  36
9 9
x4
Leave the
fraction
improper. It
must be in
lowest
terms.
Example 3
5x  16  2x  9x  15
7 x  16  9x  15
 7x
 7x
 16  2x  15
 15
 15
 1  2x
2 2
1
 x
2
Use good
form – place
the negative
out front!
Example 4 Solve.
Example 5 Solve.
32  x   5x  4x  3
 53x  2  9x  64x  5 
Example 6 Tripling the greater of two consecutive
even integers gives the same result as subtracting 10
from the lesser even integer. What are the integers?
I should write let
statements for
this word problem.
Solve.
Example 4
Example 5
32  x   5x  4x  3
 53x  2  9x  64x  5 
6  3x  5x  4x  12
 15x   10  9x  24x  30
6  2x  4x  12
 4x  4x
 6x   10  24x  30
 6x
 6x
 10  30x  30
 30
 30
20  30x
30 30
2
x
3
6  6x  12
6
6
6x  18
6
6
x  3
Example 6 Tripling the greater of two consecutive
even integers gives the same result as subtracting 10
from the lesser even integer. What are the integers?
Let f = first integer, Let f + 22 = second integer
3f  2  f  10
3f  6  f  10
f
f
2f  6  10
6 6
2f  16
2
2
f  8
82
6
The integers are -8 and -6.
Example 7 One half of a
number increased by one is
six less than one fourth of
the number. Find the
number.
1
1
n 1  n 6
2
4
2
1
n 1  n 6
4
4
1
1
 n
 n
4
4
1
n  1  6
4
1 1
 4 1
4
n


7
 
 
 1 4
1
n  28
Example 8 Sixteen more
than three fourths of a
number is one eighth of the
number less than 2. Find the
number.
3
1
n  16  2  n
4
8
6
1
n  16  2  n
8
8
1
1
 n
 n
8
8
7
n  16  2
8
 16  16
-2
 8 7 n  14 8 
 
 
 7 8
7 
n  16
Example 9 Solve. Round
to the nearest tenth.
Example 10 Solve. Round
to the nearest tenth.
4.6x  53.7  0.8x
4.5  7.2x  3.4x  49.5
10 4.6x  53.7   0.8x 10 104.5  7.2x   3.4x  49.5 10
46x  537  8x
45  72x  34x  495
 8x
 8x
 72x  72x
38x  537  0
45  106x  495
 537  537
 495
 495
38x  537
540  106x
38
38
106
106
x  14.13...You need to divide
5.09...  x
to what decimal
x  14.1 position in order to
5.1  x
round to tenths?
2-A9 Pages 101 #11–30