Let's solve the system of equations:

\[
\begin{aligned}
1.\ & 6x + 3y = 9 \\
2.\ & 5x - y = 6
\end{aligned}
\]

We can solve this using either substitution or elimination. Let's use the **elimination method** this time.

### Step 1: Simplify the first equation, if possible.
The first equation can be simplified by dividing everything by 3:
\[
2x + y = 3
\]
So the system becomes:
\[
\begin{aligned}
1.\ & 2x + y = 3 \\
2.\ & 5x - y = 6
\end{aligned}
\]

### Step 2: Add the two equations to eliminate \(y\):
Add the two equations together:
\[
(2x + y) + (5x - y) = 3 + 6
\]
\[
7x = 9
\]
\[
x = \frac{9}{7}
\]

### Step 3: Substitute \(x = \frac{9}{7}\) back into one of the original equations.
Substitute into the simplified first equation:
\[
2\left(\frac{9}{7}\right) + y = 3
\]
\[
\frac{18}{7} + y = 3
\]

### Step 4: Solve for \(y\):
Subtract \(\frac{18}{7}\) from both sides:
\[
y = 3 - \frac{18}{7}
\]
Convert 3 to a fraction with a denominator of 7:
\[
y = \frac{21}{7} - \frac{18}{7}
\]
\[
y = \frac{3}{7}
\]

### Final Answer:
The solution to the system is:
\[
x = \frac{9}{7}, \quad y = \frac{3}{7}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would you verify this solution by substituting back into the original equations?
2. How can you solve the system using the substitution method?
3. What is the graphical interpretation of this solution?
4. What is the determinant of the coefficient matrix for this system?
5. How would the solution change if the right-hand sides of the equations were different?

**Tip:** Simplifying equations before solving can make the arithmetic easier and reduce the chance of errors.

Explore the solution to the system of linear equations 6x + 3y = 9 and 5x - y = 6 using the elimination method. Understand the steps involved, including simplification and substitution. Suitable for Grades 9-12.

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