Question

the rectangle shows an array of nine numbers represented by combinations of the variables a, b, and c. answer parts (a) through (e) below.

a + b	a - b - c	a + c
a - b + c	a	a + b - c
a - c	a + b + c	a - b

d. the sum of the numbers in all rows, all columns, and the two diagonals gives eight values that are all different.
d. use the results of parts (a) through (c) to make an inductive conjecture about the rectangular array of nine numbers represented by a, b, and c
the sum of the numbers in all rows is the same value and this value is equal to 3a
the sum of the numbers in all columns is the same value and this value is equal to 3a
the sum of the numbers in the two diagonals is the same value and this value is equal to 3a
these three values are always the same
e. use deductive reasoning to prove your conjecture in part (d)
which of the following provides a deductive proof of the conjecture found in part (d)?
a. algebra can be used to add the variable expressions in the various rows, columns, and diagonals of the given array. after simplifying these sums and comparing the results, the conjecture found in part (d) is seen to be true
b. pick several other sets of 9 numbers and place them in the array and find all the appropriate sums. verify that the facts conjectured in part (d) are true. if no counterexample is found, then the conjecture is true
c. exhaustively test all combinations of values for a, b, and c and check that in all cases, the conjecture is true
d. check the values of the sums found in parts (a), (b), and (c) with a calculator and confirm that the appropriate sums have exactly the same value

Explanation:

Step1: Recall deductive reasoning concept

Deductive reasoning uses general principles to prove specific cases. In algebra, we use rules of addition and simplification of expressions.

Step2: Analyze option A

Algebraic addition of variable - expressions in rows, columns, and diagonals and simplification is a deductive method. We start with the general form of the numbers in the array (expressions in terms of a, b, and c) and use algebraic rules to find the sums. If the sums simplify to the same value (3a in this case) for all rows, columns, and diagonals, we have proven the conjecture deductively.

Step3: Analyze option B

Picking several other sets of 9 numbers is an inductive approach, not deductive. Inductive reasoning involves making generalizations based on specific examples.

Step4: Analyze option C

Exhaustively testing all combinations of values for a, b, and c is also an inductive approach. It is like checking many specific cases to support a general statement.

Step5: Analyze option D

Checking the values of the sums found in previous parts with a calculator is just verification of previous work, not a deductive proof. It doesn't use general algebraic principles to prove the conjecture.

Answer:

A. Algebra can be used to add the variable expressions in the various rows, columns, and diagonals of the given array. After simplifying these sums and comparing the results, the conjecture found in part (d) is seen to be true.