Systems of Equations with 3 Variables Coloring Activity - Algebra ... - Free Printable

It looks like you've uploaded an image of a coloring activity worksheet for Algebra, specifically focused on Systems of Equations with Three Variables. However, the actual image content (like specific equations or coloring instructions) isn't visible in this text-based interface.

But based on the description and common design of such worksheets, I can explain how these types of activities typically work and guide you through solving them.

---

🔍 What is This Activity?

This is an educational coloring worksheet where students:

1. Solve systems of three linear equations with three variables (e.g., x, y, z).
2. Use their answers to determine which color to use for certain sections of a picture (like the giraffe shown).
3. The answer key is included so that students can check their work or self-correct.

The goal is to make learning algebra more engaging by combining math with art.

---

🧩 How to Solve Systems of 3 Variables

Let’s walk through a typical example of a system of three equations:

#### Example Problem:
Solve the system:
```
(1) x + y + z = 6
(2) 2x - y + 3z = 10
(3) 3x + y - 2z = 1
```

We'll solve this using elimination.

---

✔ Step-by-Step Solution

#### Step 1: Eliminate one variable
Let’s eliminate `y` first.

Add equations (1) and (2):
```
(1) x + y + z = 6
(2) 2x - y + 3z = 10
---------------------
Add: 3x + 4z = 16 → Call this Equation (4)
```

Now add equations (1) and (3):
```
(1) x + y + z = 6
(3) 3x + y - 2z = 1
---------------------
Subtract: (3x - x) + (y - y) + (-2z - z) = 1 - 6
→ 2x - 3z = -5 → Call this Equation (5)
```

Now we have a system of two equations with two variables:
```
(4) 3x + 4z = 16
(5) 2x - 3z = -5
```

#### Step 2: Solve this 2-variable system

Multiply (4) by 3 and (5) by 4 to eliminate `z`:
```
(4)×3: 9x + 12z = 48
(5)×4: 8x - 12z = -20
```

Add:
```
(9x + 8x) + (12z - 12z) = 48 - 20
→ 17x = 28
→ x = 28/17 ≈ 1.647
```

Wait — this gives a fractional answer, which might not be ideal for a coloring sheet. Let's try another set of sample problems that yield integer solutions.

---

🟩 Better Example (with integer solutions)

Let’s use a cleaner system:

```
(1) x + y + z = 6
(2) 2x + y - z = 3
(3) x - y + 2z = 5
```

#### Step 1: Eliminate `y`

Add (1) and (2):
```
(1) x + y + z = 6
(2) 2x + y - z = 3
-------------------
Add: 3x + 2z = 9 → (4)
```

Now add (1) and (3):
```
(1) x + y + z = 6
(3) x - y + 2z = 5
-------------------
Add: 2x + 3z = 11 → (5)
```

Now solve (4) and (5):

(4) 3x + 2z = 9
(5) 2x + 3z = 11

Use elimination:

Multiply (4) by 3: 9x + 6z = 27
Multiply (5) by 2: 4x + 6z = 22

Subtract:
```
(9x - 4x) + (6z - 6z) = 27 - 22
→ 5x = 5 → x = 1
```

Plug back into (4):
3(1) + 2z = 9 → 2z = 6 → z = 3

Now plug x=1, z=3 into (1):
1 + y + 3 = 6 → y = 2

✔ Solution: x = 1, y = 2, z = 3

---

🎨 Coloring Activity Logic

Once you solve each problem, you get values like:

- If x = 1 → Color section A blue
- If y = 2 → Color section B yellow
- If z = 3 → Color section C green

Each region on the giraffe or other image corresponds to a solution value or range.

You then color accordingly to reveal a fun picture.

---

✔ Tips for Solving These Problems

1. Use elimination or substitution — elimination is usually easier.
2. Check your work by plugging the solution back into all three equations.
3. Look for patterns — sometimes one equation is a combination of others.
4. Use the answer key to verify before coloring.

---

💡 Final Thoughts

Since I can’t see the actual equations from your image, here’s what you should do:

1. Copy down each system of equations from the worksheet.
2. Solve each one using elimination/substitution.
3. Match your answers to the color code provided.
4. Color the corresponding sections of the giraffe or design.

If you’d like, you can type out the equations from the worksheet, and I’ll help you solve them step-by-step!

Would you like to do that? 😊