Transposing is an important technique for students
to master as they progress
from Algebra I to Algebra
II and beyond. It’s the process of moving quantities across the equal
sign in an equation to make the problem solving process more efficient. Here’s
how it works.

*Solving
an **equation* means isolating the variable on one side of the equation, either the right side or the
left. Generally it is better to isolate the variable
on the left side of the equal sign because we read from left to right.

Beginning algebra students are taught the **Law
of Equations**: whatever you do on one side of an equation, you must do on the other side as well.

Using the Law of Equations, to solve 2*x** *– 3 = 13, we can add
3 to both sides, giving us

2*x *– 3 + 3 = 13 + 3 ===>
2*x *= 16

Then, we divide both sides by 2
to find that *x *= 8.

To solve 2*x** *–
3 = 13 by **transposing**, we do the following:

Instead of adding 3 to both sides,
we can transpose the 3 that’s already there by

*moving it from the left side, over the equal sign, to the
right side, and changing its sign *from “–” to “+.”

Now we have 2*x *= 13 + 3, which becomes 2*x *= 16,
the same answer we got using the Law of Equations. Again, we find that *x *= 8.

The beauty of this process is that it is visually
easier to see how the parts of the equation
change as we solve it. Instead of writing the transposed quantity twice (once on the left and once on the right), we literally *pick up* the quantity and move it to the other side. The price we have to pay for doing this is changing the sign of the quantity being moved.

Let’s solve 3*y *+ 2*x *– 3 = 7 for *y*.

Since we want to
isolate *y*, we can transpose 2*x*
and – 3.

This gives us *y *= –2*x *+ 7 + 3. Simplifying,
we get *y *= –2*x *+ 10.

If we used the Law of Equations, we would have to write out

3*y *+ 2*x *– 3 = 7 ===> 3*y** *+ 2*x *– 2*x *– 3 + 3 = 7 – 2*x** *+ 3

While simplifying
the above does give us *y *= –2*x *+ 10, it comes with a lot more work.

Note that most classroom teachers will not allow transposing until students learn to use the **Law of Equations**.

*T**ry** **these:*
- Solve for
*x*: 4*x *+ 9 = 49
- Solve for
*y*: 2*y *+ 7 – 3 = 24
- Solve for
*y*: 3*x *+ 9*y *– 7 = 49
- Solve for
*a*:
3*b *– 4*a *–
6 +
4 = 49
- Solve for
*a*:
2*b *+ 4*a *+ 5 – 2 = 42
– 2*a*