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We are given a system of two equations:

1. \( 8x - y = 12 \)
2. \( 2x - 6y = 3 \)

We need to find the value of \( x + y \), where \((x, y)\) is the solution to this system.

### Step 1: Solve the system using substitution or elimination.

We will use substitution or elimination to solve for \( x \) and \( y \). First, let's solve the first equation for \( y \).

From equation (1): 
\[
8x - y = 12
\]
Rearrange to solve for \( y \):
\[
y = 8x - 12
\]

### Step 2: Substitute into the second equation

Now, substitute \( y = 8x - 12 \) into the second equation \( 2x - 6y = 3 \).

Substitute for \( y \):
\[
2x - 6(8x - 12) = 3
\]

Expand the equation:
\[
2x - 48x + 72 = 3
\]

Simplify:
\[
-46x + 72 = 3
\]

### Step 3: Solve for \( x \)

Now, solve for \( x \):
\[
-46x = 3 - 72
\]
\[
-46x = -69
\]
\[
x = \frac{-69}{-46} = \frac{69}{46} = \frac{3}{2}
\]

### Step 4: Solve for \( y \)

Substitute \( x = \frac{3}{2} \) into the equation \( y = 8x - 12 \):

\[
y = 8 \times \frac{3}{2} - 12
\]
\[
y = 12 - 12 = 0
\]

### Step 5: Find \( x + y \)

The solution to the system is \( x = \frac{3}{2} \) and \( y = 0 \). Now, calculate \( x + y \):

\[
x + y = \frac{3}{2} + 0 = \frac{3}{2}
\]

Thus, the value of \( x + y \) is \( \frac{3}{2} \).

Would you like further details or have any other questions?

Here are 5 related questions to explore further:
1. How can you solve this system using the elimination method?
2. What is the geometric interpretation of solving a system of linear equations?
3. How would the solution change if one of the equations were altered slightly?
4. What if the system had no solution or infinitely many solutions?
5. How can systems of equations be solved using matrix methods?

**Tip:** Always check your solutions by substituting back into the original equations to verify that both are satisfied.

Consider the system of equations. If (x, y) is the solution to the system, then what is the value of x + y?
1. 8x - y = 12
2. 2x - 6y = 3

Learn how to solve the system of linear equations 8x - y = 12 and 2x - 6y = 3 by substitution or elimination. This detailed step-by-step solution includes solving for x and y and determining x + y. Ideal for students in Grades 9-11.

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