Solve systems of equations using substitution in this engaging maze worksheet.

Explanation:
We are given a maze where each box contains a system of two equations. To move through the maze, we must solve each system using substitution, find the solution (x, y), and then follow the path labeled with that ordered pair to the next box — continuing until we reach “Finish”.

Let’s start at the Start box:

Start box:
System:
1) $ x = -2y $
2) $ 4x + 3y = 10 $

Substitute equation (1) into (2):
$ 4(-2y) + 3y = 10 $
→ $ -8y + 3y = 10 $
→ $ -5y = 10 $
→ $ y = -2 $

Then $ x = -2y = -2(-2) = 4 $

So solution is $ (x, y) = (4, -2) $

Now look for the arrow labeled (4, –2) leading from Start. There is one going down-right to the box with:
> $ y = 4x + 15 $
> $ -5x - y = 3 $

Let’s solve this system.

From first equation: $ y = 4x + 15 $. Substitute into second:
$ -5x - (4x + 15) = 3 $
→ $ -5x - 4x - 15 = 3 $
→ $ -9x = 18 $
→ $ x = -2 $

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

Solution: $ (-2, 7) $

Find arrow labeled (–2, 7) — it goes down to:

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

Substitute second into first:
$ -2x + (-5x - 9) = 12 $
→ $ -7x - 9 = 12 $
→ $ -7x = 21 $
→ $ x = -3 $

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

Wait — that gives $ (-3, 6) $, but let’s double-check:
Hold on — the box says:
“–2x + y = 12”
“y = –5x – 9”

Plug x = –3:
Left side: –2(–3) + y = 6 + y
Set equal to 12 → y = 6
Also y = –5(–3) – 9 = 15 – 9 = 6 ✔️
So solution is $ (-3, 6) $

Now look for arrow labeled (–3, 6) — it goes down-right to:

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

Substitute y = 3x – 10 into second:
$ 3x + 4(3x - 10) = 5 $
→ $ 3x + 12x - 40 = 5 $
→ $ 15x = 45 $
→ $ x = 3 $

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

Solution: $ (3, -1) $

Arrow labeled (3, –1) goes right to:

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

Substitute x into second:
$ 3(5y + 4) - 2y = -14 $
→ $ 15y + 12 - 2y = -14 $
→ $ 13y = -26 $
→ $ y = -2 $

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

Solution: $ (-6, -2) $

Arrow labeled (–6, –2) goes right to:

> $ x = -y - 5 $
> $ x = 2y - 2 $

Set them equal (since both = x):
$ -y - 5 = 2y - 2 $
→ $ -5 + 2 = 2y + y $
→ $ -3 = 3y $
→ $ y = -1 $

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

Solution: $ (-4, -1) $

Arrow labeled (–4, –1) goes right to Finish.

✔ So the correct path is:
Start → (4, –2) → (–2, 7) → (–3, 6) → (3, –1) → (–6, –2) → (–4, –1) → Finish

All steps check out.

Final Answer:
(4, -2) → (-2, 7) → (-3, 6) → (3, -1) → (-6, -2) → (-4, -1)