Question

The sum of three numbers is 10. The sum of twice the first​ number, 4 times the second​ number, and 5 times the third number is 33. The difference between 6 times the first number and the second number is 28. Find the three numbers.

Answer 1

1. 5
2. 2
3. 3

Explanation:

The given relations allow us to write three equations in the three unknown values. We can let x, y, z represent the three numbers, in order.

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x +y +z = 10 . . . . . . . the sum of the three numbers is 10

2x +4y +5z = 33 . . . . . designated sum is 33

6x -y = 28 . . . . . . . . 6 times the first is 28 more than the second

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There are many ways to solve a system of equations like this. Perhaps one of the easiest is to enter the equations as an augmented matrix in your calculator, and let it do the work. (Some calculators will solve the equations directly.)

The three numbers are 5, 2, and 3.

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Additional comment

The third equation lets you write an expression for y in terms of x:

y = 6x -28

Substituting this into the first two equations, we get ...

x +(6x -28) +z = 10 ⇒ 7x +z = 38

2x +4(6x -28) +5z = 33 ⇒ 26x +5z = 145

Subtracting the second from 5 times the first of these equations gives ...

5(7x +z) -(26x +5z) = 5(38) -(145)

9x = 45 . . . . . the y-variable is eliminated

x = 5

Use equations from above to find z and y.

z = 38 -7x = 3

y = 6(5) -28 = 2