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23 August, 22:08

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?

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  1. 24 August, 00:04
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    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.
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