To solve the problem of determining how many snapdragons Hans will plant if he plants 29 daisies, we need to find the linear relationship between the number of snapdragons and the number of daisies based on the given data points.
Given data points:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Number of } & \text{Number of } \\
\text{snapdragons, } x & \text{daisies, } y \\
\hline
11 & 34 \\
\hline
12 & 33 \\
\hline
13 & 32 \\
\hline
14 & 31 \\
\hline
\end{array}
\][/tex]
The relationship between the number of snapdragons ([tex]\(x\)[/tex]) and the number of daisies ([tex]\(y\)[/tex]) can be modeled with a linear equation:
[tex]\[
y = mx + b
\][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
By analyzing the data points, we calculate the slope [tex]\(m\)[/tex] and intercept [tex]\(b\)[/tex]:
[tex]\[
m = -1.0
\][/tex]
[tex]\[
b = 45.0
\][/tex]
Thus, the linear equation modeling the scenario is:
[tex]\[
y = -1.0x + 45.0
\][/tex]
To find the number of snapdragons ([tex]\(x\)[/tex]) when Hans plants 29 daisies ([tex]\(y = 29\)[/tex]), we rewrite the equation and solve for [tex]\(x\)[/tex]:
[tex]\[
29 = -1.0x + 45.0
\][/tex]
Subtract 29 from both sides:
[tex]\[
0 = -1.0x + 16.0
\][/tex]
Add [tex]\(1.0x\)[/tex] to both sides:
[tex]\[
1.0x = 16.0
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 16.0
\][/tex]
Thus, if Hans plants 29 daisies, he will plant 16 snapdragons.
The equation that models the scenario is:
[tex]\[
y = -1.0x + 45.0
\][/tex]
Hans will plant 16 snapdragons.
