Given the table of values showing the number of snapdragons ([tex]\(x\)[/tex]) and daisies ([tex]\(y\)[/tex]), we observe the following pairs:
- When [tex]\(x = 11\)[/tex], [tex]\(y = 34\)[/tex]
- When [tex]\(x = 12\)[/tex], [tex]\(y = 33\)[/tex]
- When [tex]\(x = 13\)[/tex], [tex]\(y = 32\)[/tex]
- When [tex]\(x = 14\)[/tex], [tex]\(y = 31\)[/tex]
From these pairs, we can see a pattern. As [tex]\(x\)[/tex] increases by 1, [tex]\(y\)[/tex] decreases by 1. This suggests a linear relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
To find this relationship, note that each rise of 1 in [tex]\(x\)[/tex] corresponds with a decrease of 1 in [tex]\(y\)[/tex]. We hypothesize that this relationship can be expressed as:
[tex]\[ y = c - x \][/tex]
Using any given pair to find [tex]\(c\)[/tex], let's use [tex]\((x, y) = (11, 34)\)[/tex]:
[tex]\[ 34 = c - 11 \][/tex]
Solving for [tex]\(c\)[/tex]:
[tex]\[ c = 34 + 11 = 45 \][/tex]
Thus, the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] can be modeled by the equation:
[tex]\[ y = 45 - x \][/tex]
We need to find the number of snapdragons ([tex]\(x\)[/tex]) when Hans plants 29 daisies ([tex]\(y = 29\)[/tex]):
[tex]\[ 29 = 45 - x \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = 45 - 29 \][/tex]
[tex]\[ x = 16 \][/tex]
Therefore, the equation that models this scenario is:
[tex]\[ y = 45 - x \][/tex]
And Hans will plant:
[tex]\[ x = 16 \][/tex]
In summary:
The equation [tex]\( y = 45 - x \)[/tex] models the scenario. Hans will plant 16 snapdragons.
