Algebra Lesson  Transposition
This may be your first algebra lesson. In case you are rather uncomfortable with algebra, you may want to first go through
Basic Algebra  An Introduction
 that would give you a good foundation before this lesson.
This Algebra Lesson introduces a technique known as 'transposition'. This is the most common way to solve algebra equations. A quick review here of the basic principles  all equations have two sides: a Left Side (LS) and a Right Side (RS). In the example below 3x + 4 is on the Left Side of the equation and 31 is on the Right Side of the equation:
3x + 4 = 31
LS
RS
The common transposition method is to do the same thing
(mathematically) to
both sides of the equation, with the aim of bringing like terms together and
isolating the variable (or the unknown quantity).
So, to solve this equation, first subtract 4 from both sides
of the equation. This will get rid of number 4 from the LS
3x + 4  4 = 31  4
That will give us:
3x = 27
Now, looking at the LS we have 3x. So we need to divide it by 3 to isolate x, and we need to do the same to the RS.
Now that gives us:
Check Our Answer:
Now, we have x = 9. We can substitute this value in the original equation to check if our answer is correct:
3x + 4 = 31
3 × 9 + 4 = 31
27 + 4 = 31
31 = 31
So, our answer x=9 is correct.
That's the algebra lesson on transposition. Now, you are ready to review some examples to further develop your understanding of transposition.
Algebra Lesson  Transposition: Some Examples
Example 1
x + 6 = 14
x + 6 – 6 = 14 – 6
x = 14 – 6
x = 8

Note:
Since 6 is with a + sign on the LS, we subtract 6 from both sides.
+6 and –6 cancel each other on the LS.
Check Answer:
x + 6 = 14
8 + 6 = 14

Example 2
x – 3 = 6
x – 3 + 3 = 6 + 3
x = 6 + 3
x = 9

Note:
Since 3 is with a – sign on the LS, we add 3 to both sides.
–3 and +3 cancel each other on the LS.
Check Answer:
x – 3 = 6
9 – 3 = 6

Example 3:
4x = 16
x = 4

Note:
Since 4 is multiplying x on the LS, we divide both sides by 4. On the LS
4x divided by 4 gives leaves x.
Check Answer:
4x = 16
4 x 4 = 16

Example 4:
x = 12 X
5
x = 60

Note:
Since 5 is dividing x on the LS, we multiply both sides by 5. So on the
LS 5 divided by 5 is cancelled out leaving x.
Check Answer:

Example 5:
3x + 4 = 34 – 2x
3x + 4 + 2x = 34 – 2x + 2x
5x + 4 = 34
5x + 4 – 4 = 34 – 4
5x = 30
x = 6

Note:
Since there is a –2x on the RS
we add a +2x on both sides. On the RS –2x and +2x cancel each other
out.
Since there is a +4 on the LS we subtract 4 from both sides. On the LS
+4 and –4 cancel each other out.
Since 5 multiplies x on the LS we divide both sides with 5. On the LS 5 divided by
5 cancel each other leaving x.
Check Answer:
3x + 4 = 34 – 2x
3 x 6 + 4 = 34 – 2 x
6
18 + 4 = 34 – 12
22 = 22

Example 6:
3 X (3x+4)
+ 2 X (2x+1) =
24
9x + 12 + 4x + 2 = 24
12x + 14 = 24
12x + 14 – 14 = 24 – 14
12x = 10

Note:
This equation involves fractions and we should try to get rid of
fractions first. We do this by multiplying each term with 6 (lowest
common multiple for 2 and 3 is 6)
2 cancels out 6 by 3 in the first term and 3 cancels out 6 by 2
in the second term leaving 3(3x+4) + 2(2x+1)=24
Collecting like terms together we get 12x+14=24
Since there is a +14 on the LS, we subtract 14 from both sides. On the
LS the +14 cancels out –14.
Since 12 multiplies x on the LS we divide both sides by 12. On LS 12 divided by 12 cancels out each other, leaving x equal to 10
divided by 12.
Check Answer:
You can do the math by substituting
into the original equation.

Algebra Lesson  Transposition: Practice Questions
We shall be adding Practice Questions in the near future.
Having completed the practice questions, you are now ready for other algebra lessons, like Substitution.
Useful Links:
More Algebra Lessons at Math.com
More Algebra Lessons on A Math Refresher
