Given the system of equations:

\[
\begin{cases}
3x - 18y = 15 \\
6y + 5 = x
\end{cases}
\]

Let's solve for \(x\) and \(y\) step-by-step.

### Step 1: Substitute \(x\) from the second equation into the first equation

From the second equation:

\[
x = 6y + 5
\]

Substitute this value of \(x\) into the first equation:

\[
3(6y + 5) - 18y = 15
\]

### Step 2: Simplify the equation

Expand and simplify:

\[
18y + 15 - 18y = 15
\]

Simplify further:

\[
15 = 15
\]

This is a true statement, meaning the equations are dependent (they represent the same line). Thus, there are infinitely many solutions along the line where the equations intersect.

### Step 3: General Solution

Since the equations are dependent, let’s express \(x\) and \(y\) parametrically:

Let \(y = t\), where \(t\) is any real number. Then:

\[
x = 6t + 5
\]

### Final Solution:

The general solution is:

\[
(x, y) = (6t + 5, t) \quad \text{for any real number } t.
\]

Would you like more details, or do you have any questions?

### Related Questions:

1. How do you find the intersection of two dependent lines?
2. What is a parametric solution in a system of linear equations?
3. How can you determine if two equations are dependent?
4. What does it mean geometrically when a system has infinitely many solutions?
5. How do you solve a system of linear equations with substitution?

**Tip:** In a system of linear equations, if simplification leads to a true statement like \(0 = 0\) or \(c = c\), it indicates infinitely many solutions.

Discover how to solve the system of linear equations 3x - 18y = 15 and 6y + 5 = x using substitution. This step-by-step guide explains key concepts such as parametric solutions and dependent systems with infinitely many solutions. Suitable for Grades 8-10 students.

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