What is that total? Solve for each letter.?

Question

In this magic square, A, B, and C are single-digit numbers, 2A means twice A, etc. Every row, column, and both diagonals add up to the same total. What is that total? Solve for each letter. (4 answers)

2A C 2C

A+2B A+B A

B 3A 2B

the numbers are supposed to be lined up! :)

Answer 1

The sum of the first column is 2A + (A + 2B) + B = 3A + 3B.

The last two rows also have this sum.

So let's try to figure out the middle column:

C + (A+B) + 3A = 3A + 3B

That tells us A + B + C = 3B

And look at the last column:
2C + A + 2B = 3A + 3B

Add one more A to both sides:
2C + 2A + 2B = 4A + 3B

Notice how the left side is now:
2(A + B + C) = 4A + 3B

So we can replace that with 2(3B):
2(3B) = 4A + 3B
3B = 4A

Okay, we are starting to get somewhere.

Using the prior row total of 3A + 3B, we know that each row adds to 7A:

So the first row will be:
2A + C + 2C = 7A

C + 2C = 5A
3C = 5A

Here C must be 5 and A must be 3.

And if A is 3, then B is 4.

A = 3, B = 4, C = 5
Row total is 21

Magic square:
.6 .5 10
11 7 .3
.4 .9 .8

As a double-check, all the rows, columns and diagonals add up to 21.

Answer 2

Write some equations

2A + C + 2C = D
A + 2B+ A + B + A = D
2C + A + 2B = D
2A + A +B = D

Four equations and four unknowns, solve for a, b, and c.