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The sum of three numbers is 18

The sum of three numbers is 18. The sum of twice the first number, 3 times the second number, and 4 times the third number is 53. The difference between 5 times the first number and the second number is 30.

Find the three numbers.

Number: 

Solution:

Let:
x = 1st number
y = 2nd number
z = 3rd number

The sum of three numbers is 18.
x + y + z = 18 (Eq. 1)

The sum of twice the first number, 3 times the second number, and 4 times the third number is 53.
2x + 3y + 4z = 53 (Eq. 2)

The difference between 5 times the first number and the second number is 30.
5x - y = 30 (Eq. 3)

Thus, we have three linear equations in 3 unknowns.

Eliminating Z in Eq 1 and 2, we need to multiply Eq 1 by -4 and add to Eq 2.

-4(x + y + z = 18)
-4x -4y -4z = -72 (Eq. 4)

Add Eq. 4 to Eq. 2 gives:

-4x -4y -4z = -72
2x + 3y + 4z = 53
------------------
-2x - y = -19 (Eq. 5)

Solving for x using Eq. 3 and 5:

-2x - y = -19 (Eq. 5) [Multiply by +1]

2x + y = 19
5x - y = 30 (Eq. 3)
===========
7x = 49
x = 7 (1st number)

Solving for y using Eq. 3, gives:
5x - y = 30 (Eq. 3)
5(7) - y = 30
35 - y = 30
35 - 30 = y
5 = y (2nd number)

Solving for z using Eq. 1, gives:
x + y + z = 18 (Eq. 1)
7 + 5 + z = 18
12 + z = 18
z = 18 - 12
z = 6 (3rd number)

Therefore, the 3 numbers are 7, 5, and 6.

CHECK:
x + y + z = 18 (Eq. 1)
5 + 6 + 7 = 18
18 = 18 (Check)

2x + 3y + 4z = 53 (Eq. 2)
2(7) + 3(5) + 4(6) = 53
14 + 15 + 24 = 53
53 = 53 (Check)

5x - y = 30 (Eq. 3)
5(7) - 5 = 30
35 - 5 = 30
30 = 30 (Check)


Solution:Let:x = 1st numbery

Solution:

Let:
x = 1st number
y = 2nd number
z = 3rd number

The sum of three numbers is 18.
x + y + z = 18 (Eq. 1)

The sum of twice the first number, 3 times the second number, and 4 times the third number is 53.
2x + 3y + 4z = 53 (Eq. 2)

The difference between 5 times the first number and the second number is 30.
5x - y = 30 (Eq. 3)

Thus, we have three linear equations in 3 unknowns.

Eliminating Z in Eq 1 and 2, we need to multiply Eq 1 by -4 and add to Eq 2.

-4(x + y + z = 18)
-4x -4y -4z = -72 (Eq. 4)

Add Eq. 4 to Eq. 2 gives:

-4x -4y -4z = -72
2x + 3y + 4z = 53
------------------
-2x - y = -19 (Eq. 5)

Solving for x using Eq. 3 and 5:

-2x - y = -19 (Eq. 5) [Multiply by +1]

2x + y = 19
5x - y = 30 (Eq. 3)
===========
7x = 49
x = 7 (1st number)

Solving for y using Eq. 3, gives:
5x - y = 30 (Eq. 3)
5(7) - y = 30
35 - y = 30
35 - 30 = y
5 = y (2nd number)

Solving for z using Eq. 1, gives:
x + y + z = 18 (Eq. 1)
7 + 5 + z = 18
12 + z = 18
z = 18 - 12
z = 6 (3rd number)

Therefore, the 3 numbers are 7, 5, and 6.

CHECK:
x + y + z = 18 (Eq. 1)
5 + 6 + 7 = 18
18 = 18 (Check)

2x + 3y + 4z = 53 (Eq. 2)
2(7) + 3(5) + 4(6) = 53
14 + 15 + 24 = 53
53 = 53 (Check)

5x - y = 30 (Eq. 3)
5(7) - 5 = 30
35 - 5 = 30
30 = 30 (Check)