The sum of three numbers is 91. the third number is 2 times the first. the first number is 7 less than the second. what are the numbers?

Let's call the three numbers a, b, and c.

Now we can turn the information we are given into equations.

The sum of the three numbers is 26:

a + b + c = 26

Twice the first (2 times a) minus the second (2 times a minus b) is 2 less than the third:

2a - b = c - 2

The third is the second minus three times the first:

c = b - 3a

Counting what we have here, we now have three equations and three variables: enough to solve the whole system of equations.

The third equation gives us c directly, so we can start there and substitute into the second equation:

2a - b = (b - 3a) - 2

2a + 3a = b + b - 2

5a = 2b - 2

Let's get one of these variables on its own so we can continue with the substitution:

5a + 2 = 2b

b = (5a + 2) / 2

Now we have c in terms of a and b, and b in terms of just a. So let's use the first equation and substitute to find out what a is:

a + b + c = 26

a + (5a + 2) / 2 + (b - 3a) = 26

a + (5/2) a + 1 + (5a + 2) / 2 - 3a = 26

7/2a + 1 + 5/2a + 1 - 3a = 26

12/2a + 2 - 3a = 26

6a - 3a = 26 - 2

3a = 24

a = 8

At last, we have solved for one of the variables. Now, plug this into the equation for b to find b:

b = (5a + 2) / 2 = (5 (8) + 2) / 2 = (40 + 2) / 2 = 42 / 2 = 21

Now we have a and b. Time to find c!

a + b + c = 26

(8) + (21) + c = 26

29 + c = 26

c = 26 - 29

c = - 3

So our values for a, b, and c are 8, 21, and - 3.