The system of linear equations to solve is:

$5x + 4y = 8 \quad \text{(Equation 1)}$ $10x - 4y = 46 \quad \text{(Equation 2)}$

Step 1: Add the equations

To eliminate $y$, add both equations:

$(5x + 4y) + (10x - 4y) = 8 + 46$

Simplifying:

$5x + 10x + 4y - 4y = 54$

$15x = 54$

Step 2: Solve for $x$

Now, solve for $x$ by dividing both sides by 15:

$x = \frac{54}{15} = \frac{18}{5}$

Step 3: Substitute $x$ into one of the original equations

Substitute $x = \frac{18}{5}$ into Equation 1:

$5\left(\frac{18}{5}\right) + 4y = 8$

This simplifies to:

$18 + 4y = 8$

Step 4: Solve for $y$

Now, solve for $y$:

$4y = 8 - 18$

$4y = -10$

$y = \frac{-10}{4} = \frac{-5}{2}$

Final Answer:

The solution to the system is:

$x = \frac{18}{5}, \quad y = \frac{-5}{2}$

Would you like further details or have any questions?

Here are some related questions to consider:

1. How would the method change if you wanted to use substitution instead of addition?
2. What are the advantages of solving by elimination in this case?
3. Could the solution have been found using matrices? If so, how?
4. What does the geometric interpretation of this solution look like?
5. How would you verify the solution?

Tip: When working with systems of equations, always check your solutions by substituting back into the original equations.