To determine the correct equations for this scenario, let’s break down the information given:
1. You want to buy a total of 6 potted rose plants.
2. The plants in 8-inch pots cost [tex]$12 each.
3. The plants in 10-inch pots cost $[/tex]15 each.
4. You spend a total of $78.
Let:
- [tex]\( x \)[/tex] be the number of 8-inch pots.
- [tex]\( y \)[/tex] be the number of 10-inch pots.
### Step-by-Step Solution:
1. Equating the total number of pots:
You are purchasing a total of 6 plants which gives us the equation:
[tex]\[
x + y = 6
\][/tex]
2. Equating the total cost:
The total cost is the sum of the cost of 8-inch pots and the cost of 10-inch pots. Therefore, the equation for the total cost is:
[tex]\[
12x + 15y = 78
\][/tex]
### Given Options:
1. [tex]\( x + y = 78 \)[/tex] and [tex]\( 12x + 15y = 6(78) \)[/tex]
2. [tex]\( y + 15y = 6 \)[/tex]
3. [tex]\( 12x + x = 78 \)[/tex]
4. [tex]\( x + y = 6 \)[/tex] and [tex]\( 12x + 15y = 78 \)[/tex]
5. [tex]\( x + y = 6 \)[/tex] and [tex]\( 15x + 12y = 78 \)[/tex]
### Explanation of Each Option:
- Option 1: Incorrect because [tex]\( x + y = 78 \)[/tex] is not the correct equation for the number of pots.
- Option 2: Incorrect because [tex]\( y + 15y = 6 \)[/tex] does not make logical sense.
- Option 3: Incorrect because [tex]\( 12x + x = 78 \)[/tex] is not relevant to the problem.
- Option 4: Correct, as it matches both correct equations derived from the given information.
- Option 5: Incorrect because the cost equation [tex]\( 15x + 12y = 78 \)[/tex] has the coefficients swapped.
### Conclusion:
The correct set of equations representing this scenario is:
[tex]\[
\begin{array}{l}
x + y = 6 \\
12x + 15y = 78
\end{array}
\][/tex]
Thus, the correct choice is:
[tex]\[
\boxed{4}
\][/tex]
