100+ new middle school math worksheets - Education.com Blog - Free Printable

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 our answer is (4, -2), so we look for the box with (4, -2) as an output.

Wait — actually, the solution (4, -2) corresponds to the next box that has this point as its label.

Looking at the diagram:

From Start, the arrow goes to the box with (4, -2), which is not directly labeled, but it’s the output of the first system.

Wait — let's recheck the layout.

Actually, looking at the diagram:

- The Start box has two arrows: one going down-left to (-4, -2) and one going right to (2, 4)? No.

Wait — better to analyze carefully.

Let me map the structure.

The Start box has:
- Arrow to (2, 4) → next box: `x = 3y + 8`, `x + 4y = 15`
- Arrow to (-4, -2) → next box: `y = 4x + 15`, `-5x - y = 3`

But our solution was (4, -2) — not listed as a direct node?

Wait — perhaps I made a mistake.

Let’s double-check the Start box:

> Start:
> x = -2y
> 4x + 3y = 10

We solved:
- x = -2y
- Plug in: 4(-2y) + 3y = 10 → -8y + 3y = -5y = 10 → y = -2 → x = -2(-2) = 4

So solution: (4, -2)

Now, look at the diagram: Is there a box with (4, -2) as a label?

Yes! There is a box labeled (4, -2), which is connected from Start via a line.

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

But wait — the boxes are labeled with their solutions, not their equations.

So each box has:
- A system of equations
- And a label like (2,4), (1,11), etc., which is the solution

So when we solve a system, we get a point, and we follow the arrow to the box labeled with that point.

So now, we need to find which box has the solution (4, -2).

Look at the boxes:

- Top row: (2,4), (1,11), (-1,7), (-6,1)
- Second row: (-12,3), (-2,7), (2,13), (0,3), (-1,8)
- Third row: (-4,-12), (-6,-2), (9,4), (7,-4)
- Bottom row: (3,-1), (-6,2), (-4,-1)

Wait — (4, -2) is not listed?

That can't be.

Wait — perhaps I miscalculated.

Wait — let’s check the Start box again.

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

Substitute:
4(-2y) + 3y = 10 → -8y + 3y = -5y = 10 → y = -2 → x = -2(-2) = 4 → (4, -2)

But (4, -2) is not among the labels.

But look — (4, -2) is on the edge between boxes.

Ah! Wait — the labels on the lines are the solutions, and they connect the boxes.

So the boxes have systems, and the arrows are labeled with the solution of the system.

So you solve the system in the current box, get a solution, then follow the arrow labeled with that solution to the next box.

So:

- Start box: solve → get (4, -2)
- Look for the arrow labeled (4, -2) — it leads to the next box.

From the diagram:
- From Start, there is an arrow labeled (4, -2) going to the box with:
> y = 4x + 15
> -5x - y = 3

So we go to that box.

Now solve that system.

---

🔵 Box 1:

y = 4x + 15
-5x - y = 3

Substitute y = 4x + 15 into second equation:

-5x - (4x + 15) = 3
-5x - 4x - 15 = 3
-9x = 18
x = -2

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

✔ Solution: (-2, 7)

Now follow the arrow labeled (-2, 7)

From the diagram, the arrow from this box labeled (-2, 7) goes to the box:
> -2x + y = 12
> y = -5x - 9

So go to that box.

---

🔵 Box 2:

-2x + y = 12
y = -5x - 9

Substitute y into first equation:

-2x + (-5x - 9) = 12
-7x - 9 = 12
-7x = 21
x = -3

Then y = -5(-3) - 9 = 15 - 9 = 6

✔ Solution: (-3, 6)

Follow arrow labeled (-3, 6)

It leads to:
> y = 3x - 10
> 3x + 4y = 5

Go to that box.

---

🔵 Box 3:

y = 3x - 10
3x + 4y = 5

Substitute y:

3x + 4(3x - 10) = 5
3x + 12x - 40 = 5
15x = 45
x = 3

Then y = 3(3) - 10 = 9 - 10 = -1

✔ Solution: (3, -1)

Follow arrow labeled (3, -1)

It leads to:
> x = 5y + 4
> 3x - 2y = -14

Go to that box.

---

🔵 Box 4:

x = 5y + 4
3x - 2y = -14

Substitute x:

3(5y + 4) - 2y = -14
15y + 12 - 2y = -14
13y = -26
y = -2

Then x = 5(-2) + 4 = -10 + 4 = -6

✔ Solution: (-6, 2)

Follow arrow labeled (-6, 2)

It leads to:
> x = -y - 5
> x = 2y - 2

Go to that box.

---

🔵 Box 5:

x = -y - 5
x = 2y - 2

Set equal:
-y - 5 = 2y - 2
-5 + 2 = 2y + y
-3 = 3y
y = -1

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

✔ Solution: (-4, -1)

Follow arrow labeled (-4, -1)

It leads to the Finish box!

🎉 We reached Finish!

---

✔ Final Path:

Let’s list the boxes we went through:

1. Start → Solve → (4, -2) → go to box with y = 4x + 15, -5x - y = 3
2. That box → Solve → (-2, 7) → go to box with -2x + y = 12, y = -5x - 9
3. That box → Solve → (-3, 6) → go to box with y = 3x - 10, 3x + 4y = 5
4. That box → Solve → (3, -1) → go to box with x = 5y + 4, 3x - 2y = -14
5. That box → Solve → (-6, 2) → go to box with x = -y - 5, x = 2y - 2
6. That box → Solve → (-4, -1) → go to Finish

---

✔ Answer: The correct path is:

Start → (4, -2) → (-2, 7) → (-3, 6) → (3, -1) → (-6, 2) → (-4, -1) → Finish

You can draw this path on the worksheet by connecting these points.

---

✔ Summary of Solutions:

| Box | System | Solution |
|-----|--------|----------|
| Start | x = -2y, 4x + 3y = 10 | (4, -2) |
| 1 | y = 4x + 15, -5x - y = 3 | (-2, 7) |
| 2 | -2x + y = 12, y = -5x - 9 | (-3, 6) |
| 3 | y = 3x - 10, 3x + 4y = 5 | (3, -1) |
| 4 | x = 5y + 4, 3x - 2y = -14 | (-6, 2) |
| 5 | x = -y - 5, x = 2y - 2 | (-4, -1) |
| Finish | Reached! | ✔ |

---

🎯 Final Answer:

The correct path from Start to Finish is determined by solving each system using substitution and following the solution to the next box. The sequence of solutions is:

(4, -2) → (-2, 7) → (-3, 6) → (3, -1) → (-6, 2) → (-4, -1)

Draw this path on the maze to complete the activity.