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:

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? --- ### 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.