Fill in the blanks to make the equation true. (Each blank should contain an integer.)
n/k = ? *(n - 2)/(k - 2) + ?*(n - 2)/(k - 1) + ?*(n - 2)/k
n/k = 1*(n - 2)/(k - 2) + 1*(n - 2)/(k - 1) + 1*(n - 2)/k
To find the values that should go in the blanks to make the equation true, we need some more information. Specifically, we need an equation or condition that the expression on the right-hand side of the equation satisfies.
Without any further information, the best we can do is simplify the expression on the right-hand side by combining like terms. Here's how:
1. Distribute the fractions:
(n - 2)/(k - 2) = (n - 2)/(k - 2) * 1 = (n - 2)/(k - 2) * (k - 1)/(k - 1) = (n - 2)(k - 1)/[(k - 2)(k - 1)]
(n - 2)/(k - 1) = (n - 2)/(k - 1) * 1 = (n - 2)/(k - 1) * (k - 2)/(k - 2) = (n - 2)(k - 2)/[(k - 1)(k - 2)]
(n - 2)/k = (n - 2)/k * 1 = (n - 2)/k * (k - 1)/(k - 1) = (n - 2)(k - 1)/[k(k - 1)]
2. Combine the fractions:
(n - 2)(k - 1)/[(k - 2)(k - 1)] + (n - 2)(k - 2)/[(k - 1)(k - 2)] + (n - 2)(k - 1)/[k(k - 1)]
= (n - 2)(k - 1)/[(k - 1)(k - 2)] + (n - 2)(k - 2)/[(k - 1)(k - 2)] + (n - 2)(k - 1)/[k(k - 1)]
3. Combine like terms in the numerator:
= [(n - 2)(k - 1) + (n - 2)(k - 2) + (n - 2)(k - 1)]/[(k - 1)(k - 2)]
4. Simplify:
= [(k - 1)(n - 2) + (k - 2)(n - 2) + (k - 1)(n - 2)]/[(k - 1)(k - 2)]
= [(k - 1 + k - 2 + k - 1)(n - 2)]/[(k - 1)(k - 2)]
= (3k - 4)(n - 2)/[(k - 1)(k - 2)]
So, our simplified expression is (3k - 4)(n - 2)/[(k - 1)(k - 2)].
However, without any additional information or constraints, it is not possible to find specific values to fill in the blanks to make the equation true.
To fill in the blanks in the equation, we need to determine the values for the three question marks (?).
Looking at the equation:
n/k = ? *(n - 2)/(k - 2) + ?*(n - 2)/(k - 1) + ?*(n - 2)/k
Let's simplify the equation step by step:
1. Multiply both sides of the equation by k to get rid of the denominators:
n = ? *(n - 2)(k/k - 2/k) + ?*(n - 2)(k/k - 1/k) + ?*(n - 2)(k/k)
2. Simplify the fractions:
n = ? * (n - 2) + ? * (n - 2)(k - 2)/k + ? * (n - 2)(k - 1)/k
At this point, we can see that the terms on the right side of the equation have a common factor of (n - 2). Let's factor it out:
n = (n - 2)( ? + ? * (k - 2)/k + ? * (k - 1)/k)
Now, we can match the coefficients on both sides of the equation. The coefficient of (n - 2) on the right side must be equal to 1, as in:
n = (n - 2)(1)
Therefore, in order to balance the equation, the values of the question marks must be:
? = 1
? * (k - 2)/k = 0
? * (k - 1)/k = 0
To make these equations true, we can set ? = 0, which implies:
? * (k - 2)/k = 0 * (k - 2)/k = 0
? * (k - 1)/k = 0 * (k - 1)/k = 0
Thus, the values for the three question marks to make the equation true are:
? = 1
? = 0
? = 0
The filled-in equation becomes:
n/k = 1 *(n - 2)/(k - 2) + 0*(n - 2)/(k - 1) + 0*(n - 2)/k