You are given the system of linear equations:

1. \( x + 2y = 5 \)
2. \( 3x - 15 = -6y \)

Let's solve this system step by step.

### Step 1: Simplify the second equation
We can simplify the second equation \( 3x - 15 = -6y \) by dividing the entire equation by 3 to make it easier to work with:

\[
\frac{3x - 15}{3} = \frac{-6y}{3}
\]

This simplifies to:

\[
x - 5 = -2y
\]

Or, rewriting it in a more useful form:

\[
x = -2y + 5
\]

### Step 2: Substitute into the first equation
Now that we have \( x = -2y + 5 \), we can substitute this expression for \( x \) into the first equation \( x + 2y = 5 \):

\[
(-2y + 5) + 2y = 5
\]

Simplify the equation:

\[
-2y + 5 + 2y = 5
\]

The \( -2y \) and \( 2y \) cancel each other out, leaving:

\[
5 = 5
\]

### Step 3: Interpretation
Since \( 5 = 5 \) is always true and there are no variables left, this indicates that the system of equations is **dependent**. That means the two equations represent the same line, and there are infinitely many solutions.

### Final Answer:
The system has infinitely many solutions. The solutions can be expressed in terms of \( y \):

\[
x = -2y + 5
\]

Thus, the solutions are of the form \( (x, y) = (-2y + 5, y) \), where \( y \) can be any real number.

---

Would you like further details on the solution, or have any questions?

Here are 5 related questions to explore further:
1. How can we tell if a system of equations is dependent from the start?
2. What does it mean geometrically when a system has infinitely many solutions?
3. Can all systems of two linear equations have infinitely many solutions?
4. How can we represent the solution set on a graph?
5. How would the solution change if we had different constants in these equations?

**Tip:** To check if two equations are dependent, see if one can be transformed into the other by multiplying or dividing by a constant.

Explore how to solve a system of linear equations with x + 2y = 5 and 3x - 15 = -6y. Learn about dependent systems, algebraic manipulations, and consistent solutions, suitable for Grades 8-10. The problem covers key algebra concepts and demonstrates a step-by-step approach to solving a system with infinitely many solutions.

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