I am having trouble finding a pattern or linear equation for this table

x y
1 -1
3 3
5 7
10 ?
50 ?
100 ?

Could someone help me please?

Of course! To find a pattern or linear equation for the given table, you can start by looking for a relationship between the values in the x-column and the values in the y-column.

Let's examine the differences between consecutive x-values and y-values:

For the x-values:
2 (3 - 1) is the difference between 1 and 3.
2 (5 - 3) is the difference between 3 and 5.
5 (10 - 5) is the difference between 5 and 10.
40 (50 - 10) is the difference between 10 and 50.
50 (100 - 50) is the difference between 50 and 100.

For the y-values:
4 (3 - (-1)) is the difference between -1 and 3.
4 (7 - 3) is the difference between 3 and 7.
4 (? - 7) is the difference between 7 and the next missing value.
40 (? - ?) is the difference between the two missing values.
50 (? - ?) is the difference between the two missing values.

Based on these observations, it seems that the differences in the x-values are constant (2, 2, 5, 40, 50), while the differences in the y-values are also constant (4, 4, 4, 40, 50).

Now, let's use this information to find the missing values:

Starting from the last provided y-value (7) and adding the constant difference of 4, we get:
7 + 4 = 11 (first missing value)

Continuing from 11 and adding the constant difference of 40, we get:
11 + 40 = 51 (second missing value)

Therefore, the missing values in the table are:
10 -> 11
50 -> 51
100 -> ?

Since the constant difference for the x-values is increasing (2, 5, 40, 50), it suggests that the relationship is not linear. Therefore, we need to investigate alternative patterns to find the missing value for x = 100.

One pattern that emerges is that the constant difference in the y-values is equal to the square of the constant difference in the x-values. Let's test this pattern:

Constant difference in x-values = 50 - 10 = 40
Constant difference in y-values = 40

Squaring the constant difference in the x-values:
40 * 40 = 1600

If we add this squared constant difference to the last provided y-value (7):
7 + 1600 = 1607

So, the missing value for x = 100 is y = 1607.

In summary, the completed table is as follows:

x y
1 -1
3 3
5 7
10 11
50 51
100 1607

The pattern that relates the x-values and y-values in this table is not a linear equation but appears to have a quadratic relationship.