https://file.aiquickdraw.com/mathbot/file/1728441429661a2918l4v.jpg

The system of equations provided is:

\[
\begin{aligned}
1) \quad 5x + 5y = 15 \\
2) \quad 14y - 28x = 42
\end{aligned}
\]

We need to determine how many solutions \((x, y)\) this system has. Here's the step-by-step process:

### Step 1: Simplify both equations
- First equation: \( 5x + 5y = 15 \) can be simplified by dividing through by 5:
  \[
  x + y = 3 \quad \text{(Equation 1)}
  \]
  
- Second equation: \( 14y - 28x = 42 \) can be simplified by dividing through by 14:
  \[
  y - 2x = 3 \quad \text{(Equation 2)}
  \]

### Step 2: Solve the system
Now we solve this simplified system of equations:
\[
\begin{aligned}
1) \quad x + y = 3 \\
2) \quad y - 2x = 3
\end{aligned}
\]

- From the first equation: \( y = 3 - x \).
- Substitute this into the second equation: 
  \[
  (3 - x) - 2x = 3
  \]
  Simplify:
  \[
  3 - x - 2x = 3 \quad \Rightarrow \quad 3 - 3x = 3
  \]
  Subtract 3 from both sides:
  \[
  -3x = 0 \quad \Rightarrow \quad x = 0
  \]

- Substitute \( x = 0 \) into the first equation \( x + y = 3 \):
  \[
  0 + y = 3 \quad \Rightarrow \quad y = 3
  \]

Thus, the system has exactly **one solution**, which is \( (x, y) = (0, 3) \).

### Conclusion:
The system of equations has **exactly one solution**, so the correct answer is **B: Exactly One**.

Would you like any further details or explanations?

Here are 5 related questions to expand on this topic:
1. What are the different types of solutions a system of linear equations can have?
2. How do you solve a system of equations using substitution versus elimination?
3. What is the geometric interpretation of a system with exactly one solution?
4. How do we recognize when a system of equations has no solution?
5. Can you explain how to verify the solution of a system of equations?

**Tip:** Always simplify equations as much as possible before solving—it makes the problem easier to handle!

How many solutions (x, y) does the given system of equations have?

Explore how to determine the number of solutions for the system of linear equations 5x + 5y = 15 and 14y - 28x = 42. This problem covers algebraic techniques and concepts for solving systems of equations, suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation