Daniel takes pleasure in experimenting with various kinds of toys. He bought three distinct types of toys for $25 in total. Furthermore, he possesses a toy that costs $20 more than three times the combined price of the other two toys. It's revealed that the total cost of the three toys multiplied by five amounts to $125.
Using the above scenario, answer the following questions by using the necessary steps: 
(i) Represent this situation in a system of linear equations. 
(ii) Identify the nature of the system (consistent, independent, and the number of solutions).  


Let’s break down the given problem step-by-step to answer each part.

### Given Information:
1. Daniel bought three distinct types of toys for $25 in total.
2. He has one toy that costs $20 more than three times the combined price of the other two toys.
3. The total cost of the three toys, multiplied by five, is $125.

### Define Variables:
Let:
- $ x $: cost of the first toy
- $ y $: cost of the second toy
- $ z $: cost of the third toy

### Step-by-Step Solution:

#### (i) Represent this situation in a system of linear equations.

Using the given information, we can set up the following equations:

1. **Equation 1** (Total cost of three toys):  
   Since the total cost of the three toys is $25, we have:
   \[
   x + y + z = 25
   \]

2. **Equation 2** (Cost of one toy in relation to others):  
   One toy costs $20 more than three times the combined price of the other two toys. We will assume that this statement applies to the third toy $ z $. Then:
   \[
   z = 3(x + y) + 20
   \]

3. **Equation 3** (Total cost of three toys multiplied by five):  
   We are told that the total cost of the three toys multiplied by five is $125. Thus:
   \[
   5(x + y + z) = 125
   \]

This equation can be simplified by dividing both sides by 5:
\[
x + y + z = 25
\]
This is the same as Equation 1, so it doesn't provide any new information.

Thus, our system of equations is:
\[
\begin{cases}
x + y + z = 25 \\
z = 3(x + y) + 20
\end{cases}
\]

#### (ii) Identify the nature of the system.

- **Consistency**: This system is consistent because it represents a real-world scenario with defined relationships between the variables.
- **Independence**: The system has two equations with three variables, meaning it is **dependent** and may have infinitely many solutions in terms of $ x $, $ y $, and $ z $.
- **Number of Solutions**: Since we have two equations with three variables, there are infinitely many solutions along a line or plane defined by the equations.

### Summary of Answers:
- **(i)** The system of equations representing the situation is:
  \[
  \begin{cases}
  x + y + z = 25 \\
  z = 3(x + y) + 20
  \end{cases}
  \]
- **(ii)** The system is **consistent and dependent**, with **infinitely many solutions** for $ x $, $ y $, and $ z $.

Do you want further details on solving for specific solutions or any additional help with these concepts?

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### Relative Questions
1. How would the equations change if the third toy cost $10 more instead of $20 more?
2. How can we interpret the nature of a system with more equations than variables?
3. What methods can solve a system of equations with three variables?
4. Can dependent systems have a unique solution under certain conditions?
5. How does changing the coefficient in a term affect the nature of a system?

#### Tip:
When analyzing systems of linear equations, always check for redundancy in information—it helps clarify the system's consistency and dependence.

Daniel takes pleasure in experimenting with various kinds of toys. He bought three distinct types of toys for $25 in total. Furthermore, he possesses a toy that costs $20 more than three times the combined price of the other two toys. It's revealed that the total cost of the three toys multiplied by five amounts to $125. Using the above scenario, answer the following questions by using the necessary steps:
(i) Represent this situation in a system of linear equations.
(ii) Identify the nature of the system (consistent, independent, and the number of solutions).

Explore how to represent a toy cost problem as a system of linear equations, analyzing its consistency and dependency. This problem applies algebraic methods and concepts suitable for Grades 9-11, including system of equations and dependency analysis.

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