Algebra puzzle with equations and solutions arranged in a hexagonal pattern.

Let’s solve each equation one by one. We’ll go triangle by triangle, solving for the variable (x or y) in each.

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Top-left triangle:
Equation: 7 = 2x + 19
Subtract 19 from both sides:
7 - 19 = 2x → -12 = 2x
Divide by 2: x = -6
But wait — the answer written is “x = -12” and “x = 4”. That doesn’t match. Let me check again.

Wait — maybe I misread. The triangle says:

“7 = 2x + 19” → that gives x = -6
But also says “x = -12” and “x = 4” — those are probably answers to other equations? Actually, looking at the whole figure, it seems like each triangle has an equation and its solution written inside. So we need to verify which ones are correct.

Actually, let’s treat this as a puzzle: each triangle contains an equation and possibly its solution. Our job is to solve each equation and see if the given solution matches — or perhaps just solve them all correctly.

Let me list all equations clearly:

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Triangle 1 (top left):
Equation: 7 = 2x + 19
→ 2x = 7 - 19 = -12
→ x = -6
But written solutions: x = -12, x = 4 → neither matches. Hmm.

Wait — maybe the triangle has multiple things? Looking again:

Actually, the top-left diamond-shaped section has:
- Left part: “7 = 2x + 19”
- Right part: “x = -12”, “x = 4”

That doesn’t make sense. Maybe it's two separate triangles?

Looking at the full image structure: It’s made of 8 small triangles around a center point.

Let me label them clockwise starting from top:

1. Top triangle: “7 = 2x + 19” and below it “x = 1/17”? No — actually, let’s read carefully.

Better approach: List every distinct equation shown and solve it.

From the image (reading text):

Left side vertical:
- “3y + 5 = 2” → solve for y
- “11x + 4 = 9” → solve for x

Bottom left:
- “2x + 1 = 0” → solve for x
- “x + 2 = x + 1” → solve for x

Bottom right:
- “x + 7 = 5” → solve for x
- “-x = 10” → solve for x

Right side vertical:
- “x² = 144” → solve for x
- “2x + 3 = 4x - 1” → solve for x

Top right:
- “x = 25” — that’s already solved?
Wait no — “x = 25” is written next to “2x + 3 = 4x - 1”? Not sure.

Actually, let’s extract each unique equation:

List of equations visible:

1. 7 = 2x + 19
2. x = 1/17 ← this looks like a solution, not an equation to solve
3. x = -12
4. x = 4
5. 2x + 3 = 4x - 1
6. x = -7
7. x = 25
8. x² = 144
9. x = 9
10. x + 7 = 5
11. -x = 10
12. x = -1/2
13. y = -1/2
14. 3x + 10 = 3
15. 5x - 1 = 10
16. x + 2 = x + 1
17. 2x + 1 = 0
18. 11x + 4 = 9
19. 3y + 5 = 2

Some of these are already solved (like x=4), so likely the task is to solve the unsolved equations and match or verify.

But since the user said “solve the problem accurately”, and the image shows many equations with some answers, perhaps the goal is to solve each equation that is presented as an equation (not already solved).

Let me pick only the ones that are actual equations needing solution:

Equations to solve:

A. 7 = 2x + 19
B. 2x + 3 = 4x - 1
C. x² = 144
D. x + 7 = 5
E. -x = 10
F. 3x + 10 = 3
G. 5x - 1 = 10
H. x + 2 = x + 1
I. 2x + 1 = 0
J. 11x + 4 = 9
K. 3y + 5 = 2

Now solve each:

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A. 7 = 2x + 19
2x = 7 - 19 = -12
x = -6

B. 2x + 3 = 4x - 1
Bring variables to one side:
3 + 1 = 4x - 2x
4 = 2x
x = 2

C. x² = 144
x = √144 or x = -√144
x = 12 or x = -12

D. x + 7 = 5
x = 5 - 7 = -2

E. -x = 10
Multiply both sides by -1: x = -10

F. 3x + 10 = 3
3x = 3 - 10 = -7
x = -7/3

G. 5x - 1 = 10
5x = 10 + 1 = 11
x = 11/5 = 2.2

H. x + 2 = x + 1
Subtract x from both sides: 2 = 1 → contradiction!
No solution.

I. 2x + 1 = 0
2x = -1
x = -1/2

J. 11x + 4 = 9
11x = 9 - 4 = 5
x = 5/11

K. 3y + 5 = 2
3y = 2 - 5 = -3
y = -1

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Now, comparing to what’s written in the image:

In the image, some answers are given:

- For “7 = 2x + 19”, they have “x = -12” and “x = 4” — but correct is x = -6 → so those are wrong.
- For “2x + 3 = 4x - 1”, they have “x = -7” — but correct is x = 2 → wrong.
- For “x² = 144”, they have “x = 9” — but should be ±12 → wrong.
- For “x + 7 = 5”, they have “x = -1/2” — but should be x = -2 → wrong.
- For “-x = 10”, they have “x = 10” — but should be x = -10 → wrong.
- For “3x + 10 = 3”, they have “x = 1/17” — but should be x = -7/3 → wrong.
- For “5x - 1 = 10”, they have “x = 9” — but should be x = 11/5 → wrong.
- For “x + 2 = x + 1”, no solution — they don’t show answer? Or maybe implied.
- For “2x + 1 = 0”, they have “x = -1/2” → CORRECT!
- For “11x + 4 = 9”, they have “x = 25”? Wait no — “x = 25” is elsewhere. Actually, for “11x + 4 = 9”, correct is x = 5/11, but in image it might be associated with something else.
- For “3y + 5 = 2”, they have “y = -1/2” — but correct is y = -1 → wrong.

Only one matches: “2x + 1 = 0” → x = -1/2 is correct.

But perhaps the task is simply to solve all equations, regardless of what’s written.

Since the user said “solve the problem accurately”, and didn’t specify to verify, I think we should provide correct solutions for all equations present.

But to avoid confusion, let’s assume the task is: Solve each equation shown in the diagram.

So final answers per equation:

1. 7 = 2x + 19 → x = -6
2. 2x + 3 = 4x - 1 → x = 2
3. x² = 144 → x = 12 or x = -12
4. x + 7 = 5 → x = -2
5. -x = 10 → x = -10
6. 3x + 10 = 3 → x = -7/3
7. 5x - 1 = 10 → x = 11/5
8. x + 2 = x + 1 → no solution
9. 2x + 1 = 0 → x = -1/2
10. 11x + 4 = 9 → x = 5/11
11. 3y + 5 = 2 → y = -1

Also, there are some already given like “x = 4”, “x = -12”, etc., but those seem to be incorrect attempts or distractors.

Perhaps the real task is to find which given answers are correct? But only “x = -1/2” for “2x + 1 = 0” is correct.

Alternatively, maybe the diagram is a matching game — match equation to solution.

But without more context, safest is to solve all equations correctly.

However, looking back at the image description, it seems like each triangle has an equation and its purported solution. For example:

- One triangle: “7 = 2x + 19” and “x = -12” — which is wrong.
- Another: “2x + 3 = 4x - 1” and “x = -7” — wrong.
- “x² = 144” and “x = 9” — wrong.
- “x + 7 = 5” and “x = -1/2” — wrong.
- “-x = 10” and “x = 10” — wrong.
- “3x + 10 = 3” and “x = 1/17” — wrong.
- “5x - 1 = 10” and “x = 9” — wrong.
- “x + 2 = x + 1” — no solution, but maybe they expect "no solution"
- “2x + 1 = 0” and “x = -1/2” — correct.
- “11x + 4 = 9” — what's paired? In image, “x = 25” is near “2x+3=4x-1”, not clear.
- “3y + 5 = 2” and “y = -1/2” — wrong, should be y=-1.

Also, there is “x = 4” and “x = -12” in top, which might be for other equations.

This is messy. Perhaps the intended task is to solve the equations and ignore the given answers, or to identify correct pairs.

But since the user said "solve the problem accurately", and to provide final answer, I think we should list the correct solutions for each equation that is solvable.

To simplify, let's take the equations that are clearly written as equations (not already solved) and solve them.

Final decision: Provide correct solutions for all distinct equations found in the image.

So here are the correct solutions:

- For 7 = 2x + 19: x = -6
- For 2x + 3 = 4x - 1: x = 2
- For x² = 144: x = 12 or x = -12
- For x + 7 = 5: x = -2
- For -x = 10: x = -10
- For 3x + 10 = 3: x = -7/3
- For 5x - 1 = 10: x = 11/5
- For x + 2 = x + 1: no solution
- For 2x + 1 = 0: x = -1/2
- For 11x + 4 = 9: x = 5/11
- For 3y + 5 = 2: y = -1

And the already given values like x=4, x=-12, etc., are either incorrect or for different contexts.

But perhaps the problem is to evaluate which of the given answers are correct. In that case, only "x = -1/2" for "2x + 1 = 0" is correct.

Given the ambiguity, and since the user wants accuracy, I'll provide the correct solution for each equation as listed.

For the Final Answer, since it's multiple equations, I'll list them.

But the instruction says "Final Answer: [Provide only the final correct answer]", implying one answer. That suggests perhaps the task is to find how many are correct or something else.

Re-examining the image: It's a star-like shape with 8 triangles. Each triangle has an equation and sometimes a solution written. Perhaps the task is to solve each and see if the written solution matches.

Let's do that systematically.

Label the 8 triangles (going clockwise from top):

1. Top: Equation "7 = 2x + 19", written solutions "x = -12", "x = 4" — neither correct (should be x=-6)
2. Top-right: Equation "2x + 3 = 4x - 1", written solution "x = -7" — should be x=2, so wrong
3. Right: Equation "x² = 144", written solution "x = 9" — should be ±12, so wrong
4. Bottom-right: Equation "x + 7 = 5", written solution "x = -1/2" — should be x=-2, wrong
5. Bottom: Equation "-x = 10", written solution "x = 10" — should be x=-10, wrong
6. Bottom-left: Equation "2x + 1 = 0", written solution "x = -1/2" — correct!
7. Left: Equation "11x + 4 = 9", written solution? In image, "x = 25" is nearby, but let's say for "11x+4=9", solution should be x=5/11, and if they have "x=25", wrong. Also, "3x+10=3" with "x=1/17" — should be x=-7/3, wrong.
8. Top-left: Equation "3y + 5 = 2", written solution "y = -1/2" — should be y=-1, wrong.

Also, there is "5x - 1 = 10" with "x=9" — should be x=11/5, wrong.

And "x + 2 = x + 1" — no solution, and no written answer, so perhaps not counted.

So only one correct pair: "2x + 1 = 0" with "x = -1/2".

Therefore, if the task is to identify how many given solutions are correct, the answer is 1.

If the task is to solve all, then we have multiple answers.

But given the format, and "Final Answer" singular, likely the task is to find the number of correct solutions provided in the diagram.

So let's count:

- "x = -12" for "7=2x+19" — wrong
- "x = 4" for same — wrong
- "x = -7" for "2x+3=4x-1" — wrong
- "x = 9" for "x²=144" — wrong (since 9²=81≠144)
- "x = -1/2" for "x+7=5" — wrong
- "x = 10" for "-x=10" — wrong
- "x = 25" — for what? If for "2x+3=4x-1", 2*25+3=53, 4*25-1=99, not equal — wrong
- "x = 1/17" for "3x+10=3" — 3*(1/17)+10 = 3/17 + 170/17 = 173/17 ≈10.17 ≠3 — wrong
- "x = 9" for "5x-1=10" — 5*9-1=44≠10 — wrong
- "y = -1/2" for "3y+5=2" — 3*(-1/2)+5 = -1.5+5=3.5≠2 — wrong
- "x = -1/2" for "2x+1=0" — 2*(-1/2)+1 = -1+1=0 — correct!

Also, "x = -12" is written, and for "x²=144", x=-12 is a solution, but in the diagram, "x=-12" is paired with "7=2x+19", not with "x²=144". In the top-left, "x=-12" is under "7=2x+19", so it's incorrectly paired.

Similarly, "x=12" is not written; only "x=9" for x²=144.

So only one correct pairing: "2x + 1 = 0" with "x = -1/2".

Therefore, the final answer is 1.

I think that's it.

Final Answer: 1