We are tasked with solving the system of linear equations using the elimination method:

$\text{(1)} \quad 8x - 5y = 11$ $\text{(2)} \quad 4x - 3y = 5$

Step 1: Eliminate one variable

To use elimination, we first manipulate the equations so that one variable has the same or opposite coefficient in both equations. Let's eliminate $x$ by multiplying equation (2) by 2:

$2 \times (4x - 3y) = 2 \times 5$

This gives:

$8x - 6y = 10$

Now we have the system:

$\text{(1)} \quad 8x - 5y = 11$ $\text{(3)} \quad 8x - 6y = 10$

Step 2: Subtract the equations

Next, subtract equation (3) from equation (1) to eliminate $x$:

$(8x - 5y) - (8x - 6y) = 11 - 10$

Simplifying both sides:

$8x - 5y - 8x + 6y = 1$

$y = 1$

Step 3: Substitute $y = 1$ back into one of the original equations

Now that we know $y = 1$, substitute it into either of the original equations. Let's use equation (2):

$4x - 3y = 5$

Substitute $y = 1$:

$4x - 3(1) = 5$

$4x - 3 = 5$

Add 3 to both sides:

$4x = 8$

Solve for $x$:

$x = 2$

Final Answer:

The solution to the system of equations is:

$x = 2, \quad y = 1$

Would you like more details or have any questions about this solution?

Here are 5 related questions for further practice:

1. Solve the system of equations: $3x + 2y = 7$ and $5x - y = 3$ using elimination.
2. Solve the system of equations: $4x - 3y = 6$ and $2x + y = 4$ using elimination.
3. What is the elimination method and how does it compare to substitution for solving systems of equations?
4. Solve the system of equations: $x + 3y = 9$ and $2x - y = 5$ using elimination.
5. Solve the system of equations: $6x - 7y = 13$ and $2x + 3y = 10$ using elimination.

Tip: When using elimination, it's crucial to ensure that the coefficients of one variable (either $x$ or $y$) match or are opposites in both equations. This will allow you to add or subtract the equations to eliminate that variable.