16  7  8  51
11  8 17  67
9   5 13  38
10  7  3   x.       what is the value of x

Let's solve for \( x \) in the matrix:

\[
\begin{matrix}
16 & 7 & 8 & 51 \\
11 & 8 & 17 & 67 \\
9 & 5 & 13 & 38 \\
10 & 7 & 3 & x
\end{matrix}
\]

We'll check if the value of \( x \) follows any patterns based on the other rows. Notice the following:

In each row, adding the first three numbers gives the value of the fourth number. Let's verify this pattern for each row:

- For the first row: \( 16 + 7 + 8 = 31 \), but the fourth number is 51.
- For the second row: \( 11 + 8 + 17 = 36 \), but the fourth number is 67.
- For the third row: \( 9 + 5 + 13 = 27 \), but the fourth number is 38.

It looks like the sum of the first three numbers isn't directly equal to the fourth. But, if we check the differences:

- First row: \( 51 - (16 + 7 + 8) = 51 - 31 = 20 \)
- Second row: \( 67 - (11 + 8 + 17) = 67 - 36 = 31 \)
- Third row: \( 38 - (9 + 5 + 13) = 38 - 27 = 11 \)

So, for each row, the sum of the first three numbers plus a specific value gives the fourth number. These differences are 20, 31, and 11.

Now, for the fourth row:

\[
10 + 7 + 3 = 20
\]

We see that there may be an additional value that needs to be added to 20, similar to the previous rows. Let's check the pattern of the differences: 20, 31, 11. These don't follow a simple arithmetic sequence, but we can observe that 20 is the smallest, 31 is the largest, and 11 is in between.

To keep the pattern, it seems likely that the difference in this row could be \( 20 - 9 = 9 \).

Therefore, \( x = 20 + 9 = 29 \).

The value of \( x \) is \( \boxed{29} \).

---

Do you have any questions, or would you like more details on this method?

Here are five related questions for further practice:
1. Can you determine the value of \( y \) in a similar matrix with a different pattern?
2. How would the solution change if we used multiplication instead of addition?
3. Can we find a different pattern that might explain the relationship between these numbers?
4. What if we subtract the values in each row—how would that change the fourth column?
5. Can this pattern be extended to larger matrices?

**Tip:** Always look for arithmetic patterns (addition, subtraction, etc.) in problems like these.

16 7 8 51
11 8 17 67
9 5 13 38
10 7 3 x. what is the value of x?

Solve a matrix problem where each row follows an arithmetic pattern to determine the value of x. Learn how to use sum patterns and differences in rows to arrive at the solution, ideal for students in Grades 6-8.

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