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

21 41 10
(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 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
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, columns, and diagonals is the same value 72
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
the sum of the numbers in all columns is the same value and this value is equal to
the sum of the numbers in the two diagonals is the same value and this value is equal to
these three values

Explanation:

Step1: Calculate row - sums

First row: $(a + b)+(a - b - c)+(a + c)=3a$.
Second row: $(a - b + c)+a+(a + b - c)=3a$.
Third row: $(a - c)+(a + b + c)+(a - b)=3a$.

Step2: Calculate column - sums

First column: $(a + b)+(a - b + c)+(a - c)=3a$.
Second column: $(a - b - c)+a+(a + b + c)=3a$.
Third column: $(a + c)+(a + b - c)+(a - b)=3a$.

Step3: Calculate diagonal - sums

Main diagonal: $(a + b)+a+(a - b)=3a$.
Other diagonal: $(a - c)+a+(a + c)=3a$.

Answer:

The sum of the numbers in all rows, columns, and diagonals is the same value $3a$