Exercise 25 Page 118

Let's analyze each step in the reasoning carefully. We are given that a=b, and the first step is multiplying this by b using the Multiplication Property of Equality. If a=b. then a * c = b * c. Substituting c with b in this property, we can see that this step was performed correctly. a * b =b * b ⇔ ab=b^2

Next, the Subtraction Property of Equality was used as shown in the table below.

Subtraction Property of Equality	Our case
If a=b, then a -c=b -c.	ab=b^2 ⇒ ab -a^2 = b^2 -a^2

As we can see, this step was done correctly, so let's continue analyzing. The Distributive Property says that we can use multiplication to distribute a to each term of the sum or difference within the parentheses. c | c Sum & Difference a(b+c)= ab+ ac & a (b-c)= ab- ac The equation becomes a(b-a)=(b+a)(b-a). Let's distribute to check if it was done correctly.

a(b-a)=(b+a)(b-a)

Distr

Distribute a

ab-a^2=(b+a)(b-a)

Distr

Distribute (b+a)

ab-a^2= (b+a)b-(b+a)a

Distr

Distribute b & a

ab-a^2 = b^2+ab-ab-a^2

SubTerm

Subtract term

ab-a^2=b^2-a^2

We got the previous equation after distributing, so it was done correctly! Next we are given that the Division Property of Equality was used. Let's review what it says. Ifa=b and c ≠ 0, then a c = b c. Notice that there are two conditions that have to be satisfied in order to use this property. The first one is that a=b. In our case this condition is satisfied. a(b-a)=(b+a)(b-a) The second one, however, is c ≠ 0. We can tell that the equation was divided (b-a), so it must be different than 0. Unfortunately, this is not the case. a=b ⇔ b-a=0 The expression by which the equation was divided is equal to . That is the error, and that is why this reasoning is incorrect.