The problem asks to find the value of $y$ in the following system of linear equations:

$5x - y = 8$ $x + 4y = 10$

We can solve this system of equations using substitution or elimination. Let's proceed with the substitution method.

Step 1: Solve the first equation for $y$

From the first equation, $5x - y = 8$, we can isolate $y$:

$y = 5x - 8$

Step 2: Substitute this expression for $y$ into the second equation

Substitute $y = 5x - 8$ into the second equation $x + 4y = 10$:

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

Simplify the equation:

$x + 20x - 32 = 10$ $21x - 32 = 10$ $21x = 42$ $x = 2$

Step 3: Substitute $x = 2$ back into the expression for $y$

Now substitute $x = 2$ into $y = 5x - 8$:

$y = 5(2) - 8 = 10 - 8 = 2$

Thus, the value of $y$ is 2.

Would you like more details on any step, or have further questions?

Related Questions:

1. How can we solve the system using the elimination method?
2. What happens if you have a system with no solution or infinitely many solutions?
3. How can the graphical representation of these two equations help in understanding the solution?
4. What is the general approach to solving linear systems with more than two variables?
5. Can this method be used to solve nonlinear systems of equations?

Tip: Always double-check your final answers by substituting both values of $x$ and $y$ back into the original equations.