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

12 21 6
(simplify your answers.)
what do you observe? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
(simplify your answers.)
a. the sum of the numbers in all rows, columns, and diagonals is the same value 39
b. the sum of the numbers in all rows and columns is the same value. the sum of the numbers in the two diagonals is the same value. these two values are not the same
c. the sum of the numbers in all rows is the same value. the sum of the numbers in all columns is the same value. the sum of the numbers in the two diagonals is the same value. these three values are not the same
d. the sum of the numbers in all rows, all columns, and the two diagonals gives eight values that are all different.
b. repeat part (a) for a = 15, b = 6, and c = 2
find the array obtained when a = 15, b = 6, and c = 2. some of the entries have already been filled in. fill in the remaining entries.

21	7	17
11
	23

(simplify your answers.)

Explanation:

Step1: Calculate array elements

Given $a = 15$, $b=6$, $c = 2$.
For the second - row, second - element which is $a$: Substitute $a = 15$.
For the second - row, third - element which is $a + b - c$: $15+6 - 2=19$.
For the third - row, first - element which is $a - c$: $15-2 = 13$.
For the third - row, third - element which is $a - b$: $15 - 6=9$.

Step2: Calculate sums of rows, columns and diagonals

Row 1 sum: $21+7 + 17=45$.
Row 2 sum: $11+15+19 = 45$.
Row 3 sum: $13+23+9=45$.
Column 1 sum: $21 + 11+13=45$.
Column 2 sum: $7+15 + 23=45$.
Column 3 sum: $17+19+9=45$.
Diagonal 1 (top - left to bottom - right) sum: $21+15+9=45$.
Diagonal 2 (top - right to bottom - left) sum: $17+15+13=45$.

Answer:

The completed array is:

21	7	17
11	15	19
13	23	9

The sum of the numbers in all rows, columns, and diagonals is the same value 45. So the answer for the observation part (similar to part a) is A. The sum of the numbers in all rows, columns, and diagonals is the same value 45.