Let's analyze the given data:
| Tickets purchased | Entries |
|-------------------|---------|
| 1 | 3 |
| 2 | 4 |
| 3 | 5 |
| 4 | 6 |
We notice that as the number of tickets increases, the number of entries also increases. We want to find the linear relationship between the number of tickets [tex]\(x\)[/tex] and the number of entries [tex]\(y\)[/tex].
First, we calculate the change in the number of entries as the number of tickets increases. From the table:
- When the number of tickets increases from 1 to 2, the entries increase from 3 to 4:
[tex]\[
\Delta \text{entries} = 4 - 3 = 1
\][/tex]
- When the number of tickets increases from 2 to 3, the entries increase from 4 to 5:
[tex]\[
\Delta \text{entries} = 5 - 4 = 1
\][/tex]
- When the number of tickets increases from 3 to 4, the entries increase from 5 to 6:
[tex]\[
\Delta \text{entries} = 6 - 5 = 1
\][/tex]
Overall, the change in entries ([tex]\(\Delta \text{entries}\)[/tex]) is consistently 1 for every one ticket increase ([tex]\(\Delta \text{tickets} = 1\)[/tex]).
From this, we can write the slope [tex]\(m\)[/tex] of the linear relationship [tex]\(y = mx + b\)[/tex] as:
[tex]\[
m = \frac{\Delta \text{entries}}{\Delta \text{tickets}} = \frac{1}{1} = 1
\][/tex]
Next, we find the y-intercept [tex]\(b\)[/tex]. Using the point (1, 3) from the table and substituting it into our linear equation [tex]\( y = mx + b \)[/tex], we get:
[tex]\[
3 = 1 \cdot 1 + b \implies b = 3 - 1 = 2
\][/tex]
So, the linear equation that represents the relationship between the number of tickets [tex]\(x\)[/tex] and the number of entries [tex]\(y\)[/tex] is:
[tex]\[
y = 1x + 2 \implies y = x + 2
\][/tex]
Now, we need to determine how many entries there will be if 20 tickets are purchased. Substitute [tex]\(x = 20\)[/tex] into the equation:
[tex]\[
y = 20 + 2 = 22
\][/tex]
Therefore, if 20 tickets are purchased, there will be [tex]\(\boxed{22}\)[/tex] entries in the raffle drawing.
