Why Do Signs Change in Algebra? (Article 58)

In article 58 of Algebra I, we read the following rule:

1. Write the quantity to be subtracted under that from which it is to be taken, placing similar terms under each other.
2. Conceive the signs of all the terms of the subtrahend to be changed, and then reduce by the rule for Addition.

Step 2 of this rule leads to some confusion. Why should we “conceive the signs of all the terms of the subtrahend to be changed”?

The reason is because operations are being performed within the subtrahend before it is subtracted from the minuend in the problem. These operations have an effect on the outcome of the final subtraction.

Let us look at this example: From 3ax – 2y, take 2ax + 3y.

Here, we are subtracting one quantity from another. The first quantity is 3ax – 2y. The second quantity is 2ax + 3y. It is helpful to set these in parentheses so that we remember that they are separate quantities. Thus, from (3ax – 2y), we will take (2ax + 3y). We are not subtracting the quantities 2ax and 3y from the quantity (3ax – 2y). We are subtracting the quantity (2ax + 3y) from the quantity (3ax – 2y).

Why does this matter?

Subtracting the two quantities from the minuend will produce a different result than subtracting the result of their operation as a quantity.

Let a = 2, x = 3 and y = 4.

1. (3ax – 2y) – (2ax + 3y)
2. (18 – 8) – (12 + 12)
3. (10) – (24) = -14

Note the value of the subtrahend in line 3 and the answer, in bold.

Now, let us take the quantities of the subtrahend out from the parentheses and see how this operation is performed incorrectly:

1. (3ax – 2y) – 2ax + 3y
2. (18 – 8) – 12 + 12
3. (10) – 12 + 12 = 10

In the first operation, we subtracted 12 from the minuend and then subtracted 12 more, leaving a difference of -14. In the second operation, we subtracted 12 from the minuend, but then added 12 back! This is not what the problem asked us to do.

When we keep the quantities in parentheses, the solution is easy. However, when we take the quantities of the subtrahend out from the parentheses, we must change the signs to make sure that the same result is obtained.

1. (3ax – 2y) – 2ax + 3y
2. (18 – 8) – 12 + 12
3. (10) – 12 – 12 = -14

Here, we show that after subtracting 12, we are subtracting 12 again.

When the quantities of the subtrahend are removed from the parentheses and subtracted individually, the signs must be changed to make sure the same effect is had on the minuend.

Now, let us read the rule again and understand:

1. Write the quantity to be subtracted under that from which it is to be taken, placing similar terms under each other.
2. Conceive the signs of all the terms of the subtrahend to be changed, and then reduce by the rule for Addition.

I hope this is helpful.

God bless your studies,
Mr. William C. Michael
Classical Liberal Arts Academy