Certainly! Let's solve this problem step-by-step using a linear equation.
1. Understand the Problem: We need to find out after how many weeks Kimberly's plant will have the same height as Juliette's plant.
2. Defining Variables:
- Let [tex]\( w \)[/tex] represent the number of weeks.
3. Juliette’s Plant:
- Initial height of Juliette’s plant = 12 cm.
- Growth rate per week = 1 cm/week.
- Therefore, the height of Juliette's plant after [tex]\( w \)[/tex] weeks can be represented as:
[tex]\[
h_J = 12 + 1w
\][/tex]
4. Kimberly’s Plant:
- Initial height of Kimberly’s plant = 0 cm (since it starts from seed).
- Growth rate per week = 2 cm/week.
- Therefore, the height of Kimberly's plant after [tex]\( w \)[/tex] weeks can be represented as:
[tex]\[
h_K = 2w
\][/tex]
5. Setting Up the Equation:
- We need to find out when the height of Kimberly’s plant ( [tex]\( h_K \)[/tex] ) equals the height of Juliette’s plant ( [tex]\( h_J \)[/tex] ):
[tex]\[
12 + 1w = 2w
\][/tex]
6. Solving the Equation:
- Subtract [tex]\( 1w \)[/tex] from both sides to isolate the variable [tex]\( w \)[/tex] on one side:
[tex]\[
12 = 2w - 1w
\][/tex]
- Combine like terms:
[tex]\[
12 = w
\][/tex]
7. Solution:
- Therefore, it will take [tex]\( 12 \)[/tex] weeks for Kimberly’s plant to equal the height of Juliette’s plant.
In conclusion, the number of weeks it will take for Kimberly’s plant to reach the same height as Juliette's is:
[tex]\[ \boxed{12} \][/tex]
