8x-3y=2                                    -8x-5y=46

We can solve this system of linear equations using either substitution or elimination. I’ll use the **elimination method** here.

The system of equations is:

1. \( 8x - 3y = 2 \)
2. \( -8x - 5y = 46 \)

### Step 1: Add the two equations to eliminate \(x\).
\[
(8x - 3y) + (-8x - 5y) = 2 + 46
\]
This simplifies to:
\[
-8y = 48
\]

### Step 2: Solve for \(y\).
\[
y = \frac{48}{-8} = -6
\]

### Step 3: Substitute \(y = -6\) back into one of the original equations to solve for \(x\). Let’s use the first equation \(8x - 3y = 2\):
\[
8x - 3(-6) = 2
\]
\[
8x + 18 = 2
\]
\[
8x = 2 - 18
\]
\[
8x = -16
\]
\[
x = \frac{-16}{8} = -2
\]

### Final Solution:
\[
x = -2, \quad y = -6
\]

Would you like further details or have any questions?

### Relative questions:
1. How would you solve a system of equations using substitution?
2. Can we solve this system graphically and verify the solution?
3. How do we check the solution for consistency in both equations?
4. What are dependent and independent systems in linear equations?
5. How can you apply elimination to more complex systems?

**Tip:** Always verify your solution by substituting the values of \(x\) and \(y\) back into the original equations to ensure they hold true.

Learn how to solve the system of linear equations 8x - 3y = 2 and -8x - 5y = 46 using the elimination method. This step-by-step guide covers key algebraic concepts such as solving for variables using elimination, substitution, and verifying solutions. Suitable for students in Grades 8-10.

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