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Let's solve the Systems of Equations Substitution Maze step by step.

We are given a maze where each box contains a system of two equations. We must solve each system using substitution, find the solution (x, y), and use that to determine which path to follow from Start to Finish.

---

🟩 Step 1: Start at the "Start" Box

Start Box:
```
x = -2y
4x + 3y = 10
```

#### Substitute x = -2y into the second equation:
$$
4(-2y) + 3y = 10 \\
-8y + 3y = 10 \\
-5y = 10 \\
y = -2
$$

Now plug back into x = -2y:
$$
x = -2(-2) = 4
$$

✔ Solution: (4, -2)

So we go to the box labeled (4, -2)

---

🟨 Next Box: (4, -2)

This is connected to:
- (2, 4)
- (11, 1)
- (-4, -2)
- (-12, 3)

But we came from (4, -2), so look at which box has (4, -2) as its solution.

Wait — actually, the solution of the system in a box determines which arrow to follow.

So now, we need to solve the next system whose solution is (4, -2)? No.

Actually, each box has a system. The answer (x, y) tells you where to go next.

So we solved the start box → got (4, -2). That means we move to the box labeled with (4, -2).

Looking at the diagram:

From Start, the path goes to the box with coordinates (4, -2), which is the box below and to the right of Start.

That box has:
```
x = 3y + 8
x + 4y = 15
```

Let’s solve this system.

---

🔹 Box: x = 3y + 8, x + 4y = 15

Substitute x = 3y + 8 into the second equation:
$$
(3y + 8) + 4y = 15 \\
7y + 8 = 15 \\
7y = 7 \\
y = 1
$$

Now find x:
$$
x = 3(1) + 8 = 11
$$

✔ Solution: (11, 1)

So we go to the box labeled (11, 1)

---

🔹 Box: (11, 1)

Look for the box with this solution.

The box above it is:
```
y = 12x - 5
y = 3x + 4
```

Let’s solve this system.

Set the two expressions for y equal:
$$
12x - 5 = 3x + 4 \\
12x - 3x = 4 + 5 \\
9x = 9 \\
x = 1
$$

Then y = 3(1) + 4 = 7

✔ Solution: (1, 7)

So we go to the box labeled (1, 7)

---

🔹 Box: (1, 7)

Which box has this solution?

Look at the box:
```
x = -5y - 1
y = 2x + 13
```

Solve this system.

We can substitute y = 2x + 13 into the first equation:
$$
x = -5(2x + 13) - 1 \\
x = -10x - 65 - 1 \\
x = -10x - 66 \\
x + 10x = -66 \\
11x = -66 \\
x = -6
$$

Now y = 2(-6) + 13 = -12 + 13 = 1

✔ Solution: (-6, 1)

So go to (-6, 1)

---

🔹 Box: (-6, 1)

Find the box with this solution.

Look at:
```
y = 4x + 15
-5x - y = 3
```

Solve:
Substitute y = 4x + 15 into second equation:
$$
-5x - (4x + 15) = 3 \\
-5x - 4x - 15 = 3 \\
-9x = 18 \\
x = -2
$$

Then y = 4(-2) + 15 = -8 + 15 = 7

✔ Solution: (-2, 7)

Go to (-2, 7)

---

🔹 Box: (-2, 7)

Next box:
```
y = -4x - 5
3x + y = -3
```

Substitute y = -4x - 5 into second equation:
$$
3x + (-4x - 5) = -3 \\
3x - 4x - 5 = -3 \\
-x - 5 = -3 \\
-x = 2 \\
x = -2
$$

Then y = -4(-2) - 5 = 8 - 5 = 3

✔ Solution: (-2, 3)

Go to (-2, 3)

---

🔹 Box: (-2, 3)

Next box:
```
x = -y + 5
3x + 5y = 15
```

Substitute x = -y + 5 into second equation:
$$
3(-y + 5) + 5y = 15 \\
-3y + 15 + 5y = 15 \\
2y + 15 = 15 \\
2y = 0 \\
y = 0
$$

Then x = -0 + 5 = 5

✔ Solution: (5, 0)

Go to (5, 0)

---

🔹 Box: (5, 0)

Next box:
```
y = 6x + 14
2x + y = 6
```

Substitute y = 6x + 14 into second equation:
$$
2x + (6x + 14) = 6 \\
8x + 14 = 6 \\
8x = -8 \\
x = -1
$$

Then y = 6(-1) + 14 = -6 + 14 = 8

✔ Solution: (-1, 8)

Go to (-1, 8)

---

🔹 Box: (-1, 8)

Next box:
```
x = -4y - 9
3x + 6y = -3
```

Substitute x = -4y - 9 into second equation:
$$
3(-4y - 9) + 6y = -3 \\
-12y - 27 + 6y = -3 \\
-6y - 27 = -3 \\
-6y = 24 \\
y = -4
$$

Then x = -4(-4) - 9 = 16 - 9 = 7

✔ Solution: (7, -4)

Go to (7, -4)

---

🔹 Box: (7, -4)

Now, check the box labeled (7, -4) — is it near Finish?

Yes! It's one of the boxes directly connected to Finish.

Now check the box:
```
x = -y - 5
x = 2y - 2
```

Set them equal:
$$
-y - 5 = 2y - 2 \\
-5 + 2 = 2y + y \\
-3 = 3y \\
y = -1
$$

Then x = -(-1) - 5 = 1 - 5 = -4

Or x = 2(-1) - 2 = -2 - 2 = -4

✔ Solution: (-4, -1)

Go to (-4, -1)

---

🔹 Box: (-4, -1)

Now we're at (-4, -1), which is connected to Finish.

Check if this box has the correct system.

The box says:
```
x = -y - 5
x = 2y - 2
```

We already solved this — it gives (-4, -1)

And from here, there's a path to Finish.

✔ So we’ve reached Finish!

---

✔ Final Path Summary:

Let’s list the sequence of solutions we found:

1. Start: Solve → (4, -2)
2. Go to box with solution (4, -2): `x = 3y + 8`, `x + 4y = 15` → (11, 1)
3. Go to (11, 1): `y = 12x - 5`, `y = 3x + 4` → (1, 7)
4. Go to (1, 7): `x = -5y - 1`, `y = 2x + 13` → (-6, 1)
5. Go to (-6, 1): `y = 4x + 15`, `-5x - y = 3` → (-2, 7)
6. Go to (-2, 7): `y = -4x - 5`, `3x + y = -3` → (-2, 3)
7. Go to (-2, 3): `x = -y + 5`, `3x + 5y = 15` → (5, 0)
8. Go to (5, 0): `y = 6x + 14`, `2x + y = 6` → (-1, 8)
9. Go to (-1, 8): `x = -4y - 9`, `3x + 6y = -3` → (7, -4)
10. Go to (7, -4): `x = -y - 5`, `x = 2y - 2` → (-4, -1)
11. Go to (-4, -1) → Finish

---

✔ Answer: The correct path from Start to Finish is through the following solution points:

(4, -2) → (11, 1) → (1, 7) → (-6, 1) → (-2, 7) → (-2, 3) → (5, 0) → (-1, 8) → (7, -4) → (-4, -1) → Finish

You would draw lines connecting these boxes in order.

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✔ Final Check: Did we reach Finish?

Yes! The last box before Finish is (-4, -1), and it connects directly to Finish.

So the maze is successfully solved.

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🎉 Conclusion:

You solve each system using substitution, then use the solution to determine your next move. By following the correct path, you get from Start to Finish.

✔ Answer: The correct path is determined by solving each system and moving along the solution points as shown above.