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

a. determine the nine numbers in the array for a = 13, b = 7, and c = 1. what do you observe about the sum of the numbers in all rows, all columns, and the two diagonals?
fill in the array obtained when a = 13, b = 7, and c = 1.
(simplify your answers.)

Explanation:

Step1: Calculate the first - row elements

Substitute \(a = 13\), \(b = 7\), \(c = 1\) into the first - row expressions.
For \(a + b\): \(13+7=20\).
For \(a - b - c\): \(13 - 7-1=5\).
For \(a + c\): \(13 + 1=14\).

Step2: Calculate the second - row elements

For \(a - b + c\): \(13 - 7 + 1=7\).
For \(a\): \(a = 13\).
For \(a + b - c\): \(13+7 - 1=19\).

Step3: Calculate the third - row elements

For \(a - c\): \(13 - 1=12\).
For \(a + b + c\): \(13+7 + 1=21\).
For \(a - b\): \(13 - 7=6\).

Step4: Calculate row - sums

First row sum: \(20 + 5+14=39\).
Second row sum: \(7 + 13+19=39\).
Third row sum: \(12 + 21+6=39\).

Step5: Calculate column - sums

First column sum: \(20 + 7+12=39\).
Second column sum: \(5 + 13+21=39\).
Third column sum: \(14 + 19+6=39\).

Step6: Calculate diagonal - sums

First diagonal sum (\(a + b\), \(a\), \(a - b\)): \(20+13 + 6=39\).
Second diagonal sum (\(a - b - c\), \(a\), \(a + b + c\)): \(5+13+21=39\).

Answer:

7	13	19
12	21	6

The sum of the numbers in all rows, all columns, and the two diagonals is 39.