The image contains a shape equation puzzle where various shapes represent unknown values, and there are four clues given in the form of equations. Let's define each shape as follows:

• Let the triangle (△) represent a variable $x$.
• Let the square (□) represent a variable $y$.
• Let the hexagon (⎔) represent a variable $z$.

The clues provided are:

1. $x + y + z = \text{(Unknown)}$
2. $x + y = 13$
3. $x - y = 1$
4. $z + y = 7$

Step-by-Step Solution:

From Clue 2: $x + y = 13 \quad \text{(Equation 1)}$

From Clue 3: $x - y = 1 \quad \text{(Equation 2)}$

By adding Equation 1 and Equation 2: $(x + y) + (x - y) = 13 + 1$ $2x = 14$ $x = 7$

Now, substitute $x = 7$ into Equation 1: $7 + y = 13$ $y = 6$

Now, use Clue 4: $z + y = 7$ Substitute $y = 6$: $z + 6 = 7$ $z = 1$

So, the values are:

• $x = 7$ (Triangle)
• $y = 6$ (Square)
• $z = 1$ (Hexagon)

Now we can use these values to check Clue 1: $x + y + z = 7 + 6 + 1 = 14$

Thus, the solution to the puzzle is correct!

Would you like more details or have any questions? Here are some related questions to consider:

1. Can you solve another shape equation puzzle by changing the shapes?
2. How do systems of equations help solve puzzles like this?
3. What would happen if one of the clues were different?
4. How can you check if the solution is consistent across all clues?
5. How would you solve this puzzle using substitution or elimination?

Tip: When solving a system of equations, always look for clues that allow easy elimination or substitution of variables.