To solve the system of linear equations by substitution, we will go through the following steps:

Given: $x - 5y = 10$ $-2x - 6y = -4$

Step 1: Solve for $x$ in terms of $y$ in the first equation.

From the equation $x - 5y = 10$, we isolate $x$: $x = 10 + 5y$

Step 2: Substitute $x = 10 + 5y$ into the second equation.

Substitute this expression for $x$ in the second equation: $-2(10 + 5y) - 6y = -4$

Expanding this: $-20 - 10y - 6y = -4$ $-20 - 16y = -4$

Step 3: Solve for $y$.

Add 20 to both sides: $-16y = 16$ Now, divide by -16: $y = -1$

Step 4: Substitute $y = -1$ back into the equation $x = 10 + 5y$ to find $x$.

$x = 10 + 5(-1)$ $x = 10 - 5$ $x = 5$

Solution

The solution to the system is: $x = 5, \quad y = -1$

Would you like further details on the steps, or do you have any questions?

Here are some related questions for further practice:

1. How would you solve a system using the elimination method instead?
2. What if both equations were set equal to each other instead of using substitution?
3. How does the substitution method compare to graphing for solving systems?
4. Could there be a scenario where substitution is not the ideal method?
5. What happens if we rearrange both equations before starting the substitution?

Tip: In substitution, always isolate one variable first and substitute accurately to avoid calculation errors.