# 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.

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)

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