To solve the given system of equations using substitution, we proceed as follows:

Given System of Equations:

1. $y = -2x - 7$
2. $2y - x = 1$

Step 1: Substitute $y$ from Equation 1 into Equation 2

Since $y = -2x - 7$, we can replace $y$ in Equation 2 with this expression: $2(-2x - 7) - x = 1$

Step 2: Simplify the Equation

Expanding and simplifying: $-4x - 14 - x = 1$ Combine like terms: $-5x - 14 = 1$

Step 3: Solve for $x$

Add 14 to both sides: $-5x = 15$ Now, divide by -5: $x = -3$

Step 4: Substitute $x = -3$ back into Equation 1 to find $y$

Using $y = -2x - 7$: $y = -2(-3) - 7$ $y = 6 - 7$ $y = -1$

Solution

The solution to the system is: $(x, y) = (-3, -1)$

Would you like further details on any part of this process, or have any additional questions?

Here are some related questions to consider:

1. How would the process change if both equations had the same slope?
2. What is the graphical interpretation of this solution in terms of lines intersecting?
3. Could this system be solved using elimination? How would it compare to substitution?
4. What is the effect of changing the constant in either equation on the solution?
5. Can this substitution method be extended to systems of three equations?

Tip: When using substitution, always simplify expressions as much as possible to avoid calculation errors.