Consider the equations below. What is the sum of a, b, and c?

Question

3a + b = 17

5b + c = 14

3a + 5c = 41

Answer 1

Whenever there is a problem like this that doesn't directly ask for the solution set, it is sometimes beneficial to try to find the answer to their question without finding the solution set--in other words to solve for an expression, rather than for individual variables. If I were to guess the source of this question, I would guess a test like ACT, SAT, or GRE. They really love these oh-so-clever questions that have simple, elegant solutions that eliminate the need for down-and-dirty algebra.

With a little outside-the-box thinking, we find that there actually is a quick, clever, and elegant way to find a + b + c, without needing to find a, b, or c individually. I suggest rewriting the system of equations this way:

3a + b = 17

5b + c = 14

3a + 5c = 41

After a little bit of observation, you may notice that the sums of the a coefficients, the b coefficients, and the c coefficients are all equal to 6. That is a rather curious thing. So what would happen if we added all three equations together?

We would get the following equation:

6a + 6b + 6c = 72

This is eerily close to what we're trying to find. Since all the coefficients are equal to 6, we can divide both sides by 6, to get

a + b + c = 12.

Note: By this method, I still don't have values for a, b, or c. But that's ok. That's not what I was after.

Answer 2

To find the sum of a, b, and c, we’ll first find what a and c are equal to in terms of b. Our original system of equations is:

3a + b = 17

5b + c = 14

3a + 5c = 41

We’ll subtract b from both sides of the first equation which gives us 3a = 17 - b. We can then plug 17 - b in for 3a in our third equation which gives us 17 - b + 5c = 41.

Next we’ll take our second equation and subtract 5b from both sides, giving us c = 14 - 5b. We can then plug this value in for c in our third equation which gives us 17 - b + 5 (14 - 5b) = 41.

Now we’ll solve for b by distributing the 5 to everything inside the parentheses. 17 - b + 70 - 25b = 41. Next we’ll combine like terms. 87 - 26b = 41. Subtract 87 from both sides. -26b = -46. Divide both sides by -26. b = 23/13.

We can plug this into the first equation to solve for a.

3a + 23/13 = 17

3a = 17 - 23/13

3a = 198/13

a = 198/39 which can be simplified to a = 66/13

We’ll also plug b into our second equation to solve for c.

5 (23/13) + c = 14

115/13 + c = 14

c = 14 - 115/13

c = 67/13

a + b + c = ?

66/13 + 23/13 + 67/13 = 156/13 which reduces to 12 :)

a + b + c = 12