this is a 6 part question. i give out 5 star super easy so give it a try! \"Simp

Question

this is a 6 part question. i give out 5 star super easy so give it a try!

"Simplify" and the properties that define a field.

One student last year provided the following solution to the equation:

(6x + 3)*(2/3) = -10

Multiply each side by 3/2:

(3/2)((6x + 3)*(2/3)) = -10(3/2)

Simplify:

6x + 3 = -15.

Subtract 3 from each side:

(6x + 3) - 3 = -15 -3

Simplify:

6x = -18

Divide each side by 6:

6x/6 = -18/6

Simplify:

x = -3.

Here are six problems related to this solution.

1. With the first "simplify", the student replaced (3/2)((6x + 3)*(2/3)) by 6x+3. This replacement is the application of the following:

For all a, b in a field, a*(b*a^{-1}) = b.

Prove this identity using the properties of a field.

2. With the first "simplify", the student also replaced -10(3/2) by -15. This could be viewed as an application of the following:

For all a, b, c in a field, a*(b*c^{-1}} = (a*c^{-1})*b.

Prove this identity using the properties of a field.

3. When we write -10(3/2), there is an implicit assumption that the negative of any element of a field not only exists but is unique. Prove, using the properties of a field, that an element of a field cannot have more than one negative.

4. For the next "simplify" the student replaced -15 - 3 by -18. To make sense of this, we need to define subtraction, which we do by: a - b = a + (-b). Then the "simplify" is a an application of the following:

For all a, b in a field, (-a) + (-b) = -(a + b).

Prove this identity, using the properties of a field or facts previously proven in this module (or in Section 7A) using the properties of a field. (For example, you may want to use what you proved in #3.)

5. The student used "simplify" to replace (6x + 3) - 3 by 6x. This appears to be an application of the following:

For all a, b in a field, (a + b) - b = a.

Prove this identity using the properties of a field.

6. The student used "simplify" to replace 6x/6 by x: Write a general identity for all a and b in a field that is a generalization of 6x/6 = x , and prove it using the properties of a field. (Think of an expression of the form a/b as having the form a*b^{-1}, where b^{-1} is the multiplicative inverse of b in the field.)

Explanation / Answer

1. field is commutative

So, a*(b*a^{-1}) = a*(a^{-1}*b)

Associative

= (a*a^{-1})*b

= 1*b

=b

2. first apply commutative law and then associative:

a*(b*c^{-1}} =a*(c^{-1}*b) = (a*c^{-1})*b.

3. Let a be an element

let b and c be negatives of a

So, a+b = 0

Add c to right on both sides:

a + (b+c) = 0+c

a+(c+b) = c (commutative)

(a+c)+b = c (associative)

b = c ( because a+c = 0)

Hence there exists a unique negative

4.Negative of a+b = -(a+b)

So, a+b+ (-(a+b)) = 0

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

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

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

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

5.(a+b) - b

=(a+b) + (-b)

=a + (b+(-b))

= a + 0

= a

6. Identity : (a*b)*a^{-1} = b

LHS = (b*a) * a^{-1} (commutative)

= b*(a*a^{-1}) (associative)

= b*1

=b

Hence proved.

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