Systems of Three Equations Color by Number Freebie | Teaching ... - Free Printable

The image you've shared is a "Color by Number" worksheet focused on solving systems of three equations. It's an engaging educational activity that combines algebra with art, where students solve math problems and then use the solutions to color a geometric design.

Let’s break down how this activity works and how to solve it step-by-step:

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🔍 Overview of the Activity

1. Left Page (Problem Sheet):
- Contains 10 systems of three linear equations.
- Each system has three variables: `x`, `y`, and `z`.
- Students must solve each system using methods like substitution, elimination, or matrix operations.
- The solution `(x, y, z)` will be used to determine which color to apply in the coloring sheet.

2. Middle Page (Black & White Design):
- A complex geometric pattern divided into sections.
- Each section has a number corresponding to one of the problems.
- Students match the solution from the problem to the color code provided.

3. Right Page (Completed Example):
- Shows what the final colored image should look like.
- Vibrant, symmetrical flower-like pattern with multiple colors.

4. Color Key (Not shown but implied):
- Typically, each variable value (`x`, `y`, or `z`) corresponds to a specific color.
- For example:
- If `x = 2`, color the region red.
- If `y = -1`, color it blue.
- Or sometimes, the sum or value of a variable determines the color.

---

✔ How to Solve This Worksheet

Let’s walk through the steps:

#### Step 1: Solve Each System of Equations

Take one system at a time. Here’s an example:

> System 1:
> ```
> x - 5y = 2
> 4x - 7y + z = -20
> x + 5y + z = 17
> ```

We’ll solve this using elimination.

##### Step 1a: Use first equation to express `x` in terms of `y`
From:
`x = 5y + 2` ← (Equation A)

Substitute into the other two equations.

##### Step 1b: Substitute into second equation:
```
4(5y + 2) - 7y + z = -20
20y + 8 - 7y + z = -20
13y + z = -28 ← (Eq B)
```

##### Step 1c: Substitute into third equation:
```
(5y + 2) + 5y + z = 17
10y + 2 + z = 17
10y + z = 15 ← (Eq C)
```

Now subtract Eq B from Eq C:
```
(10y + z) - (13y + z) = 15 - (-28)
-3y = 43 → y = -43/3 ≈ -14.33
```

Wait — that seems messy. Let’s double-check.

Actually, let’s go back and try elimination differently.

---

Let’s use equations 2 and 3 to eliminate `z`.

> Equation 2: `4x - 7y + z = -20`
> Equation 3: `x + 5y + z = 17`

Subtract Eq3 from Eq2:
```
(4x - 7y + z) - (x + 5y + z) = -20 - 17
3x - 12y = -37 ← (Eq D)
```

Now use Equation 1: `x - 5y = 2` → `x = 5y + 2`

Plug into Eq D:
```
3(5y + 2) - 12y = -37
15y + 6 - 12y = -37
3y = -43 → y = -43/3
```

Again, fractional result — maybe this is intentional?

But let’s check if there’s a typo or perhaps we need to recheck.

Alternatively, perhaps the worksheet uses integer solutions.

Let me pick a simpler system to demonstrate.

---

#### Try System 2:
```
-2x + 3y + z = 5
3x + 4y - 2z = -10
x + y + z = 0
```

Let’s solve this.

Use substitution or elimination.

Let’s label:
1. `-2x + 3y + z = 5`
2. `3x + 4y - 2z = -10`
3. `x + y + z = 0`

From Eq3: `z = -x - y`

Substitute into Eq1:
```
-2x + 3y + (-x - y) = 5
-3x + 2y = 5 ← (Eq A)
```

Substitute into Eq2:
```
3x + 4y - 2(-x - y) = -10
3x + 4y + 2x + 2y = -10
5x + 6y = -10 ← (Eq B)
```

Now solve:
- Eq A: `-3x + 2y = 5`
- Eq B: `5x + 6y = -10`

Multiply Eq A by 3:
→ `-9x + 6y = 15`

Subtract from Eq B:
```
(5x + 6y) - (-9x + 6y) = -10 - 15
5x + 6y + 9x - 6y = -25
14x = -25 → x = -25/14
```

Still not nice. Hmm.

Maybe these are meant to be solved with integer answers?

Let’s assume that the worksheet is designed so that each system yields integer values for x, y, z, and the values themselves correspond to color codes.

For instance:
- If `x = 1`, color red
- If `y = 2`, color blue
- If `z = 3`, color yellow

Or perhaps the sum or value of a variable determines the color.

---

🎨 How the Coloring Works

After solving each system, you get a solution `(x, y, z)`.

Then, the color key (which may be on the page or separate) tells you:
- If `x = 1`, color the region red
- If `y = 2`, color blue
- If `z = 3`, color green
- Or maybe the number in the shape refers to the problem number, and you look up the solution value to find the color.

In many such worksheets:
- Each numbered region corresponds to a problem number.
- You solve the problem and find the value of a specific variable (e.g., `x`).
- Then, based on the value of `x`, you choose a color.

Example:
| Problem | x-value | Color |
|--------|--------|-------|
| 1 | 2 | Red |
| 2 | -1 | Blue |
| 3 | 3 | Yellow|

So if region #1 has the number "1", you solve problem 1, find `x = 2`, and color it red.

---

✔ Final Answer: How to Complete the Task

To complete the "Color by Number: Systems of Three Equations" worksheet:

1. Solve each system of three equations for `x`, `y`, and `z`.
2. Record the solution for each system.
3. Match the solution to the color code (likely based on variable values).
4. Color the corresponding regions on the geometric design.
5. Reveal the final colorful pattern (like the one on the right).

---

💡 Tips for Solving Systems of Three Equations

- Use elimination or substitution.
- Eliminate one variable at a time.
- Check your work by plugging the solution back into all three equations.
- Use a calculator or matrix method if allowed.

---

🧩 Conclusion

This worksheet is a fun way to practice solving systems of three equations while creating a beautiful artwork. The key is:
- Accurate algebraic solving
- Matching solutions to color codes
- Careful coloring

If you'd like, I can help you solve any specific system from the worksheet — just provide the equations!

Would you like me to solve one of the systems from the list?