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\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Number of \\
snapdragons, [tex]$x$[/tex]
\end{tabular} & \begin{tabular}{c}
Number of \\
daisies, [tex]$y$[/tex]
\end{tabular} \\
\hline 11 & 34 \\
\hline 12 & 33 \\
\hline 13 & 32 \\
\hline 14 & 31 \\
\hline
\end{tabular}

Hans is planting a garden with snapdragons and daisies. The table shows some possible combinations of the two plants. If Hans plants 29 daisies, how many snapdragons will he plant?

The equation [tex]$\square$[/tex] models the scenario.

Hans will plant [tex]$\square$[/tex] snapdragons.

Sagot :

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.