Solve algebra equations to color this intricate mandala using the provided color key.

To solve this "Color by Number" puzzle, we need to find the value of $x$ for each equation. Then, we match that number to the color key provided in the image.

The Color Key:
* $3 = \text{Blue}$
* $4 = \text{Pink}$
* $5 = \text{Purple}$
* $6 = \text{Red}$
* $7 = \text{Green}$
* $8 = \text{Orange}$
* $9 = \text{Yellow}$

Here is the step-by-step solution for every section of the mandala, starting from the center and moving outward.

Center Star

* Equation: $1x + 38 = 42$
* Solve: Subtract 38 from both sides: $1x = 4$. So, $x = 4$.
* Color: Pink (Note: The text inside says "Red", but the math proves it is Pink).

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Inner Petals (The 8 shapes surrounding the center)

*We will go clockwise, starting from the top petal.*

1. Top Petal:
* Equation: $122 + -12x = 38$
* Solve: Subtract 122 from both sides: $-12x = -84$. Divide by -12: $x = 7$.
* Color: Green

2. Top-Right Petal:
* Equation: $33 - 3x = 15$
* Solve: Subtract 33 from both sides: $-3x = -18$. Divide by -3: $x = 6$.
* Color: Red

3. Right Petal:
* Equation: $2x + 18 = 26$
* Solve: Subtract 18 from both sides: $2x = 8$. Divide by 2: $x = 4$.
* Color: Pink

4. Bottom-Right Petal:
* Equation: $18 + -2x = 12$
* Solve: Subtract 18 from both sides: $-2x = -6$. Divide by -2: $x = 3$.
* Color: Blue

5. Bottom Petal:
* Equation: $-4x = 71 = 43$ (This is written as $71 - 4x = 43$)
* Solve: Subtract 71 from both sides: $-4x = -28$. Divide by -4: $x = 7$.
* Color: Green

6. Bottom-Left Petal:
* Equation: $29 + 5x = 44$
* Solve: Subtract 29 from both sides: $5x = 15$. Divide by 5: $x = 3$.
* Color: Blue

7. Left Petal:
* Equation: $75 + 2x = 81$
* Solve: Subtract 75 from both sides: $2x = 6$. Divide by 2: $x = 3$.
* Color: Blue

8. Top-Left Petal:
* Equation: $5x - 12 = 28$
* Solve: Add 12 to both sides: $5x = 40$. Divide by 5: $x = 8$.
* Color: Orange

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Outer Ring Segments

*We will continue going clockwise around the outer edge.*

9. Top Segment:
* Equation: $60 + 2x = 74$
* Solve: Subtract 60 from both sides: $2x = 14$. Divide by 2: $x = 7$.
* Color: Green

10. Top-Right Segment:
* Equation: $88 - 5x = 53$
* Solve: Subtract 88 from both sides: $-5x = -35$. Divide by -5: $x = 7$.
* Color: Green

11. Right-Middle Segment:
* Equation: $103 - 8x = 79$
* Solve: Subtract 103 from both sides: $-8x = -24$. Divide by -8: $x = 3$.
* Color: Blue

12. Lower-Right Segment:
* Equation: $1x + 64 = 67$
* Solve: Subtract 64 from both sides: $1x = 3$. So, $x = 3$.
* Color: Blue

13. Bottom-Right Segment:
* Equation: $23 + 4x = 55$
* Solve: Subtract 23 from both sides: $4x = 32$. Divide by 4: $x = 8$.
* Color: Orange

14. Bottom Segment:
* Equation: $-4x + 49 = 29$
* Solve: Subtract 49 from both sides: $-4x = -20$. Divide by -4: $x = 5$.
* Color: Purple

15. Bottom-Left Segment:
* Equation: $31 + 4x = 55$
* Solve: Subtract 31 from both sides: $4x = 24$. Divide by 4: $x = 6$.
* Color: Red

16. Lower-Left Segment:
* Equation: $129 + -13x = 38$
* Solve: Subtract 129 from both sides: $-13x = -91$. Divide by -13: $x = 7$.
* Color: Green

17. Left-Bottom Segment:
* Equation: $32 + 4x = 60$
* Solve: Subtract 32 from both sides: $4x = 28$. Divide by 4: $x = 7$.
* Color: Green

18. Left-Middle Segment:
* Equation: $-1x + 21 = 18$
* Solve: Subtract 21 from both sides: $-1x = -3$. Divide by -1: $x = 3$.
* Color: Blue

19. Upper-Left Segment:
* Equation: $71 + 1x = 74$
* Solve: Subtract 71 from both sides: $1x = 3$. So, $x = 3$.
* Color: Blue

20. Top-Left Segment:
* Equation: $48 + 1x = 56$
* Solve: Subtract 48 from both sides: $1x = 8$. So, $x = 8$.
* Color: Orange

21. Top-Left Inner Segment:
* Equation: $120 + 7x = 85$ ... Wait, looking closely at the image, this equation is likely $120 - 7x = 85$ or similar because $120+something$ won't equal a smaller number with positive x. Let's re-read carefully. It looks like $120 + 7x = 85$ is impossible for positive integers. Let me re-examine crop 4. Ah, it is actually $120 - 7x = 85$? No, the sign is clearly `+`. Let's look at the next one.
* Actually, let's look at $39 + 3x = 54$.
* Solve: $3x = 15 \rightarrow x = 5$. Color: Purple.
* Let's look at $-9 + 4x = 15$.
* Solve: $4x = 24 \rightarrow x = 6$. Color: Red.
* Let's look at $-2x + 59 = 41$.
* Solve: $-2x = -18 \rightarrow x = 9$. Color: Yellow.
* Let's re-evaluate the segment labeled $120 + 7x = 85$. If I assume it's a typo for subtraction ($120 - 7x = 85$), then $-7x = -35$, so $x=5$ (Purple). If it is addition, there is no whole number solution. Given the pattern of other answers being whole numbers 3-9, it is highly likely a subtraction sign printed poorly or intended to be subtraction. However, looking at the very similar equation nearby $76 + -4x = 40$, the negative sign is explicit. Let's look at $48 + 1x = 56$ again. That was $x=8$.
* Let's check the segment between Purple ($x=5$) and Red ($x=6$). The equation is $120 + 7x = 85$. This is physically impossible for positive x. Let's look really closely at the original image. It might be $12 + 7x = 85$? No. It might be $120 - 7x = 85$. Let's assume Purple (5) for now based on the likelihood of a typo for minus.
* *Correction*: Looking at the segment $71 + 3x = 86$ (next to it).
* Solve: $3x = 15 \rightarrow x = 5$. Color: Purple.
* Let's check the segment $5 + 3x = 23$.
* Solve: $3x = 18 \rightarrow x = 6$. Color: Red.
* Let's check the segment $6 + 3x = 18$.
* Solve: $3x = 12 \rightarrow x = 4$. Color: Pink.
* Let's check the segment $76 + -4x = 40$.
* Solve: $-4x = -36 \rightarrow x = 9$. Color: Yellow.

Let's re-verify the tricky left-side ones:
* Equation: $120 + 7x = 85$. If this is actually $12 + 7x = 85$, $7x=73$ (no). If it is $120 - 7x = 85$, $x=5$ (Purple). Given the neighbor is Purple ($x=5$) and the other neighbor is Red ($x=6$), Purple fits the visual flow.
* Equation: $71 + 3x = 86$. $3x = 15$, $x=5$. Color: Purple.

Summary of Colors by Location

Center:
* Pink

Inner Ring (Clockwise from Top):
1. Green
2. Red
3. Pink
4. Blue
5. Green
6. Blue
7. Blue
8. Orange

Outer Ring (Clockwise from Top):
1. Green ($60+2x=74$)
2. Green ($88-5x=53$)
3. Blue ($103-8x=79$)
4. Blue ($1x+64=67$)
5. Orange ($23+4x=55$)
6. Purple ($-4x+49=29$)
7. Red ($31+4x=55$)
8. Green ($129-13x=38$)
9. Green ($32+4x=60$)
10. Blue ($-1x+21=18$)
11. Blue ($71+1x=74$)
12. Orange ($48+1x=56$)
13. Purple ($120-7x=85$ *assumed typo for minus*) OR if strictly reading plus, no solution. But wait, look at $39+3x=54$ -> $x=5$ (Purple). And $-9+4x=15$ -> $x=6$ (Red). And $-2x+59=41$ -> $x=9$ (Yellow).
* Let's re-read the segment labeled $120 + 7x = 85$. It is located between the Purple ($x=5$) and Red ($x=6$) sections on the left.
* Let's check the equation $71 + 3x = 86$. $x=5$ (Purple).
* Let's check the equation $39 + 3x = 54$. $x=5$ (Purple).
* Let's check the equation $-9 + 4x = 15$. $x=6$ (Red).
* Let's check the equation $-2x + 59 = 41$. $x=9$ (Yellow).

There seems to be a cluster on the upper left.
* Outer tip: Yellow ($x=9$)
* Next inner: Red ($x=6$)
* Next inner: Purple ($x=5$)
* Next inner: Purple ($x=5$)
* The problematic one: $120 + 7x = 85$. If we assume it's $12 + 7x = 82$? No. It is almost certainly a typo for $120 - 7x = 85$ which yields Purple (5).

Let's check the bottom-left area again.
* $76 + -4x = 40$ -> $x=9$ (Yellow).
* $6 + 3x = 18$ -> $x=4$ (Pink).
* $5 + 3x = 23$ -> $x=6$ (Red).

Final Answer:
Center: Pink
Inner Ring (starting top, going clockwise): Green, Red, Pink, Blue, Green, Blue, Blue, Orange
Outer Ring (starting top, going clockwise):
1. Green
2. Green
3. Blue
4. Blue
5. Orange
6. Purple
7. Red
8. Green
9. Green
10. Blue
11. Blue
12. Orange
13. Purple (assuming typo for subtraction in $120+7x=85$)
14. Purple
15. Red
16. Yellow