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The sum of three numbers $x$ ,$y$, $z$ is 165. When the smallest number $x$ is multiplied by 7, the result is $n$. The value $n$ is obtained by subtracting 9 from the largest number $y$. This number $n$ also results by adding 9 to the third number $z$. What is the product of the three numbers?

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Answer:

Explanation:

Let's break down the given information step by step to solve the problem:

The sum of three numbers x, y, and z is 165: x + y + z = 165.

When the smallest number x is multiplied by 7, the result is n: 7x = n.

The value n is obtained by subtracting 9 from the largest number y: y - 9 = n.

The value n is also obtained by adding 9 to the third number z: z + 9 = n.

We can use this information to form a system of equations:

Equation 1: x + y + z = 165

Equation 2: 7x = n

Equation 3: y - 9 = n

Equation 4: z + 9 = n

To find the product of the three numbers, we need to determine the values of x, y, z, and n.

First, let's solve for n using Equation 2:

7x = n

Now, let's substitute the value of n into Equations 3 and 4:

Equation 3: y - 9 = 7x

Equation 4: z + 9 = 7x

We can rearrange Equation 3 to express y in terms of x:

y = 7x + 9

Now, we can substitute the value of y in Equation 1:

x + (7x + 9) + z = 165

Simplifying:

8x + z = 156

Now, we have two equations:

Equation 4: z + 9 = 7x

8x + z = 156

We can solve this system of equations to find the values of x and z:

From Equation 4, we have z = 7x - 9.

Substituting z in Equation 8x + z = 156:

8x + (7x - 9) = 156

15x - 9 = 156

15x = 165

x = 11

Substituting x = 11 in Equation 4:

z + 9 = 7(11)

z + 9 = 77

z = 68

Now, we have the values of x = 11, y = 7x + 9 = 7(11) + 9 = 86, and z = 68.

The product of the three numbers x, y, and z is:

Product = x * y * z = 11 * 86 * 68 = 64648.

Therefore, the product of the three numbers is 64648.

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