https://file.aiquickdraw.com/mathbot/file/1726014288976kmmziipz.jpg

Let's break down the problem and create the system of equations based on the information given:

### Variables:
- Let $ x $ represent the number of transformers.
- Let $ y $ represent the number of circuit breakers.
- Let $ z $ represent the number of capacitors.

### Step 1: Writing the System of Equations
1. **Total number of electrical components:**
   There are a total of 30 components, so:
   \[
   x + y + z = 30
   \]

2. **Cost constraints:**
   The cost of each component is:
   - Transformer: \$500 each,
   - Circuit breaker: \$100 each,
   - Capacitor: \$50 each.
   
   The total budget is \$8500, so the cost equation becomes:
   \[
   500x + 100y + 50z = 8500
   \]

3. **Relationship between components:**
   The number of capacitors and circuit breakers together is equal to the number of transformers, so:
   \[
   y + z = x
   \]

### Step 2: Solving the System of Equations
We now have the following system of equations:
\[
\begin{aligned}
1. & \quad x + y + z = 30 \\
2. & \quad 500x + 100y + 50z = 8500 \\
3. & \quad y + z = x
\end{aligned}
\]

Let's solve this system. I'll compute it step by step.The solution to the system of equations is:
- $ x = 15 $ transformers,
- $ y = 5 $ circuit breakers,
- $ z = 10 $ capacitors.

Thus, for this installation, you should use 15 transformers, 5 circuit breakers, and 10 capacitors.

Do you need any further explanation or details? 

Here are five related questions you might find helpful:
1. How would the system change if the total budget were increased or decreased?
2. What happens to the number of components if the price of one component changes?
3. Can you formulate a similar system if more types of electrical components were involved?
4. How would you handle a situation where the relationship between components was different?
5. What are some practical scenarios in real life where similar systems of equations are used?

**Tip:** When solving systems of equations, it's helpful to simplify the system by substituting variables to reduce the number of unknowns step by step.

Suppose that you are working on designing an electrical network for an industrial installation. You have three types of electrical components: transformers, circuit breakers, and capacitors. You need to install a total of 30 electrical components in the network. Transformers cost $500 each, circuit breakers cost $100 each, and capacitors cost $50 each. Moreover, the total number of capacitors and circuits breakers is equal to the number of transformers. You have a total budget of $8500 for the project. Let x, y and z be respectively the numbers of transformers, circuit breakers, and capacitors. Write a system of three equations with three unknowns that described the above given. By solving the previous system, deduce the number of electrical components of each type that you must use for this installation.

Learn how to solve a system of linear equations involving transformers, circuit breakers, and capacitors for an industrial network installation. This problem covers algebraic concepts like systems of equations and budgeting constraints. Suitable for high school students.

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