https://file.aiquickdraw.com/mathbot/file/1726958423977j310i1ch.jpg

The given system of equations is:

\[
\begin{cases}
15x - 6y = 6 \\
16x + 8y = 4
\end{cases}
\]

### Step 1: Simplify Each Equation

**Equation 1:**

\[
15x - 6y = 6
\]

Divide the entire equation by 3:

\[
5x - 2y = 2
\]

**Equation 2:**

\[
16x + 8y = 4
\]

Divide the entire equation by 4:

\[
4x + 2y = 1
\]

### Step 2: Solve the System Using Elimination or Substitution

Let's add the simplified equations:

\[
(5x - 2y) + (4x + 2y) = 2 + 1
\]

\[
9x = 3
\]

\[
x = \frac{3}{9} = \frac{1}{3}
\]

Now substitute \( x = \frac{1}{3} \) into one of the simplified equations, say \( 5x - 2y = 2 \):

\[
5\left(\frac{1}{3}\right) - 2y = 2
\]

\[
\frac{5}{3} - 2y = 2
\]

Multiply through by 3:

\[
5 - 6y = 6
\]

\[
-6y = 1
\]

\[
y = -\frac{1}{6}
\]

### Conclusion:
The solution to the system is:
\[
\boxed{x = \frac{1}{3}, \ y = -\frac{1}{6}}
\]

Thus, the correct choice is **A. The solution is \( x = \frac{1}{3} \) and \( y = -\frac{1}{6} \)**.

Would you like any further details or explanations?

### Related Questions:
1. How do you determine if a system of equations is consistent or inconsistent?
2. What are the methods for solving a system of linear equations?
3. Can you graphically interpret the solution to a system of equations?
4. How can you use substitution to solve a system of equations?
5. What happens when you have infinitely many solutions in a system?

**Tip:** Always simplify the equations first to make solving easier!

Solve the system of equations: 15x - 6y = 6 and 16x + 8y = 4. Determine whether the system has a solution, or if it is inconsistent.

This problem explores how to solve a system of linear equations using both the elimination and substitution methods. The example equations are 15x - 6y = 6 and 16x + 8y = 4. Simplify the equations and determine whether there is a unique solution or if the system is inconsistent.

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