First, we have to find HCF of 657 and 963
Now, 657 = 9*63
         963 = 9*107
So, HCF(657, 963) = 9
Since HCF is expressed in the form of 657x +  963 * (-15)
So, 657x +  963 * (-15) = 9
=> 657x - 963 * 15 = 9
=> 657x - 14445 = 9
=> 657x = 14445 + 9
=> 657x = 14454
=> x = 14454/657
=> x = 22
So, the value of x is 22


I hope it's helpful for us......

Explanation:

To find the highest common factor (HCF) of two numbers, we can use the Euclidean algorithm. The Euclidean algorithm states that the HCF of two numbers can be obtained by repeatedly dividing the larger number by the smaller number and taking the remainder. This process is continued until the remainder is zero.

Step 1: Find the remainder when 963 is divided by 657.
- 963 ÷ 657 = 1 remainder 306

Step 2: Find the remainder when 657 is divided by 306.
- 657 ÷ 306 = 2 remainder 45

Step 3: Find the remainder when 306 is divided by 45.
- 306 ÷ 45 = 6 remainder 36

Step 4: Find the remainder when 45 is divided by 36.
- 45 ÷ 36 = 1 remainder 9

Step 5: Find the remainder when 36 is divided by 9.
- 36 ÷ 9 = 4 remainder 0

Since the remainder is zero, we stop the process. The last non-zero remainder, which is 9, is the HCF of 657 and 963.

Expressing HCF as 657x + 963y:

Now we need to express the HCF of 657 and 963 in the form of 657x + 963y, where x and y are integers.

Step 1: Rewriting the last non-zero remainder (HCF) as a linear combination of 657 and 963.
- 9 = 45 - 36

Step 2: Substitute the remainder 36 in the above equation.
- 9 = 45 - (306 - 45)
- 9 = 45 - 306 + 45

Step 3: Rearrange the terms.
- 9 = -306 + 2(45) + 45

Step 4: Combine like terms.
- 9 = -306 + 2(45) + 45
- 9 = -306 + 2(45) + 45
- 9 = 45 - 306 + 2(45)

Step 5: Group the terms.
- 9 = (45 - 306) + 2(45)

Step 6: Combine like terms.
- 9 = -261 + 2(45)

Step 7: Simplify.
- 9 = -261 + 90

Step 8: Combine like terms.
- 9 = -261 + 90
- 9 = -171

Therefore, the HCF of 657 and 963 can be expressed as 657(-261) + 963(90) = -9.

Find x:

To find x, we compare the coefficients of 657 in the expression -9 = 657(-261) + 963(90).
From the equation, we can see that x = -261.

Therefore, x = -261.