Math worksheet with visual equations to solve for x, including answer key and graph paper.

It looks like you've uploaded an image of a math worksheet that uses algebra tiles to help students solve for the variable $ x $ in various equations. The worksheet is divided into two columns: one with visual representations using algebra tiles, and the other with equations represented symbolically. There’s also an answer key on the right.

Let me break down how to interpret and solve these problems, then explain each step clearly.

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🔍 Understanding Algebra Tiles:

- Green rectangle: Represents $ x $
- Yellow square: Represents $ +1 $
- Red square: Represents $ -1 $
- Red rectangle: Represents $ -x $

These tiles are used to model equations visually and help students understand balancing both sides.

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Let’s go through each problem in the left column (with visuals), then verify with the symbolic version on the right.

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✔ Problem 1:

Visual:
- Left side: One green tile ($ x $) and four yellow tiles ($ +4 $)
- Right side: Six yellow tiles ($ +6 $)

So the equation is:
$$
x + 4 = 6
$$

Solve:
Subtract 4 from both sides:
$$
x = 6 - 4 = 2
$$

✔ Answer: $ x = 2 $

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✔ Problem 2:

Visual:
- Left side: One green tile ($ x $) and three red squares ($ -3 $)
- Right side: Two yellow tiles ($ +2 $)

Equation:
$$
x - 3 = 2
$$

Solve:
Add 3 to both sides:
$$
x = 2 + 3 = 5
$$

✔ Answer: $ x = 5 $

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✔ Problem 3:

Visual:
- Left side: Three green tiles ($ 3x $) and one yellow tile ($ +1 $)
- Right side: Nine red squares ($ -9 $? Wait — actually, they're red squares, which are $ -1 $, so total = $ -9 $)

Wait — let's double-check:

Actually, the right side shows 9 red squares, which represent $ -9 $. But that seems odd because we have $ 3x + 1 = -9 $? Let’s confirm.

But looking at the symbolic version (right side):
It says:
$$
3x + 1 = -9
$$

Yes — so the visual must show:
- Left: 3 green tiles ($ 3x $) and 1 yellow tile (+1)
- Right: 9 red tiles (-9)

So:
$$
3x + 1 = -9
$$

Solve:
Subtract 1:
$$
3x = -10
\Rightarrow x = -\frac{10}{3}
$$

Wait! That’s not an integer. But the answer key says $ x = -3 $. Hmm — contradiction?

Let’s check the answer key:

> 3. $ 3x + 1 = -8 $ → $ x = -3 $

Ah! So the equation is likely $ 3x + 1 = -8 $, not $ -9 $. So maybe the visual has 8 red squares, not 9?

Let’s recheck the image description:

In the visual, right side shows a 3×3 grid of red squares — that’s 9 squares.

But if the equation is $ 3x + 1 = -8 $, then the right side should be 8 red squares.

Possibility: Typo or misalignment between visual and symbolic.

But wait — the symbolic version (on the right) says:

> 3. $ 3x + 1 = -8 $

And the answer key says $ x = -3 $

Check:
$ 3(-3) + 1 = -9 + 1 = -8 $ ✔

So the correct equation is $ 3x + 1 = -8 $

Therefore, the visual must show 8 red squares, not 9.

Possibly the image is slightly off, or the red tiles are arranged in a 3×3 but only 8 filled? Or it's a mistake.

But assuming the symbolic version is correct, we use:

$$
3x + 1 = -8
\Rightarrow 3x = -9
\Rightarrow x = -3
$$

✔ Answer: $ x = -3 $

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✔ Problem 4:

Visual:
- Left: One green tile ($ x $) and two yellow tiles ($ +2 $)
- Right: One yellow tile ($ +1 $)

Equation:
$$
x + 2 = 1
$$

Solve:
Subtract 2:
$$
x = 1 - 2 = -1
$$

✔ Answer: $ x = -1 $

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Now let’s check the right-hand column (symbolic versions):

They are:

1. $ x + 4 = 6 $ → $ x = 2 $ ✔
2. $ x - 3 = 2 $ → $ x = 5 $ ✔
3. $ 3x + 1 = -8 $ → $ x = -3 $ ✔
4. $ x + 2 = 1 $ → $ x = -1 $ ✔

All match the answer key.

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🧠 Summary of How to Solve These:

1. Interpret the tiles:
- Green rectangle = $ x $
- Yellow square = $ +1 $
- Red square = $ -1 $
- Red rectangle = $ -x $

2. Write the equation based on both sides.

3. Solve algebraically:
- Isolate $ x $ by adding/subtracting constants.
- Divide if needed.

4. Check with the answer key.

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✔ Final Answers (from answer key):

| Problem | Equation | Solution |
|--------|------------------|---------|
| 1 | $ x + 4 = 6 $ | $ x = 2 $ |
| 2 | $ x - 3 = 2 $ | $ x = 5 $ |
| 3 | $ 3x + 1 = -8 $ | $ x = -3 $ |
| 4 | $ x + 2 = 1 $ | $ x = -1 $ |

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💡 Teaching Tip:

This worksheet helps students visualize algebra. For example, in Problem 3, removing 1 from both sides leaves $ 3x = -9 $, so each $ x $ must be $ -3 $.

You can use physical algebra tiles or digital tools to reinforce this concept.

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If you'd like, I can generate a printable version or create similar problems. Let me know!