To solve this system of equations using substitution, let's follow these steps:

Given Equations

1. $y = -3x + 1$
2. $y = -2x - 2$

Step 1: Substitute

Since both equations are already solved for $y$, we can set them equal to each other: $-3x + 1 = -2x - 2$

Step 2: Solve for $x$

Now, rearrange the equation to isolate $x$:

1. Add $3x$ to both sides: $1 = x - 2$
2. Add $2$ to both sides: $x = 3$

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

Let's substitute $x = 3$ into the first equation: $y = -3(3) + 1$ $y = -9 + 1$ $y = -8$

Solution

The solution to the system is: $(x, y) = (3, -8)$

Verification

To verify, substitute $x = 3$ and $y = -8$ into the second equation: $y = -2(3) - 2$ $y = -6 - 2$ $y = -8$ This confirms that the solution is correct.

---

Would you like further details or have any questions?

Related Questions

1. How do you use the substitution method for systems of equations?
2. What are alternative methods to solve systems of equations?
3. Why is it important to verify solutions in algebra?
4. How do you interpret the solution of a system geometrically?
5. What are other types of solutions possible for a system of linear equations?

Tip: Always double-check solutions by substituting them back into the original equations.