To solve the given system of equations, we need to determine the point where the height of the bean plant and the flower are equal. This involves solving the following system of equations for [tex]\( h \)[/tex] (height in centimeters) and [tex]\( t \)[/tex] (time in weeks):
[tex]\[
\begin{array}{l}
h = \frac{1}{2} t + 4 \\
h = \frac{1}{4} t + 5.5
\end{array}
\][/tex]
### Step-by-Step Solution
1. Set the equations equal to each other: Since we are looking for the point where the heights are the same, set the right-hand sides of the equations equal to each other:
[tex]\[
\frac{1}{2} t + 4 = \frac{1}{4} t + 5.5
\][/tex]
2. Solve for [tex]\( t \)[/tex]: Rearrange the variables and solve for [tex]\( t \)[/tex]:
[tex]\[
\frac{1}{2} t - \frac{1}{4} t = 5.5 - 4
\][/tex]
Combine like terms:
[tex]\[
\left( \frac{1}{2} - \frac{1}{4} \right)t = 1.5
\][/tex]
Simplify the coefficient of [tex]\( t \)[/tex]:
[tex]\[
\frac{2}{4} t - \frac{1}{4} t = 1.5 \quad \Rightarrow \quad \frac{1}{4} t = 1.5
\][/tex]
Multiply both sides by 4 to solve for [tex]\( t \)[/tex]:
[tex]\[
t = 1.5 \times 4 \quad \Rightarrow \quad t = 6.0
\][/tex]
3. Substitute [tex]\( t \)[/tex] back into one of the original equations: To find the height at [tex]\( t = 6.0 \)[/tex] weeks, substitute [tex]\( t = 6.0 \)[/tex] into either of the original equations. We will use [tex]\( h = \frac{1}{2} t + 4 \)[/tex]:
[tex]\[
h = \frac{1}{2} \times 6 + 4
\][/tex]
Calculate the height:
[tex]\[
h = 3 + 4 \quad \Rightarrow \quad h = 7.0 \, \text{cm}
\][/tex]
### Interpret the Solution
The solution to the system [tex]\( (t, h) = (6.0, 7.0) \)[/tex] means that after 6 weeks, both the bean plant and the flower will be 7 centimeters tall.
So the solution is:
[tex]\[
\begin{aligned}
&\text{The solution to the system, } t = 6.0 \text{ means that the plants will both be} \\
&7.0 \, \text{centimeters tall after} \\
&6.0 \, \text{weeks}.
\end{aligned}
\][/tex]
