To solve the problem step by step, we need to find the values of the digits (1st, 2nd, 3rd, and 4th digits) based on the given equations.

The equations are:

1. $1^{st} \, \text{digit} + 2^{nd} \, \text{digit} = 6$
2. $4^{th} \, \text{digit} + 3^{rd} \, \text{digit} = 15$
3. $2^{nd} \, \text{digit} + 4^{th} \, \text{digit} = 8$
4. $1^{st} \, \text{digit} + 4^{th} \, \text{digit} = 14$
5. $3^{rd} \, \text{digit} + 1^{st} \, \text{digit} = 13$

Let the digits be represented as $x, y, z, w$ where:

• $x = 1^{st} \, \text{digit}$
• $y = 2^{nd} \, \text{digit}$
• $z = 3^{rd} \, \text{digit}$
• $w = 4^{th} \, \text{digit}$

Step-by-Step Solution

1. From Equation 1:
   $x + y = 6 \quad \text{(Equation A)}$

2. From Equation 3:
   $y + w = 8 \quad \text{(Equation B)}$

3. From Equation 4:
   $x + w = 14 \quad \text{(Equation C)}$

4. From Equation 5:
   $z + x = 13 \quad \text{(Equation D)}$

5. From Equation 2:
   $w + z = 15 \quad \text{(Equation E)}$

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Solving for the digits:

• From Equation C $(x + w = 14)$ and Equation A $(x + y = 6)$:
  Subtract $x + y = 6$ from $x + w = 14$:
  $(x + w) - (x + y) = 14 - 6$ $w - y = 8 \quad \text{(Equation F)}$ So:
  $w = y + 8$

• Substitute $w = y + 8$ into Equation B $(y + w = 8)$:
  $y + (y + 8) = 8$ $2y + 8 = 8$ $2y = 0 \quad \Rightarrow \quad y = 0$

• Now that $y = 0$, substitute $y$ into $x + y = 6$ (Equation A):
  $x + 0 = 6$ $x = 6$

• Substitute $y = 0$ into $w = y + 8$ (Equation F):
  $w = 0 + 8$ $w = 8$

• Finally, substitute $x = 6$ into $z + x = 13$ (Equation D):
  $z + 6 = 13$ $z = 7$

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Solution:

The digits are:

• $1^{st} \, \text{digit} = 6$
• $2^{nd} \, \text{digit} = 0$
• $3^{rd} \, \text{digit} = 7$
• $4^{th} \, \text{digit} = 8$

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Verification:

1. $x + y = 6 \quad \Rightarrow \quad 6 + 0 = 6$ ✔
2. $w + z = 15 \quad \Rightarrow \quad 8 + 7 = 15$ ✔
3. $y + w = 8 \quad \Rightarrow \quad 0 + 8 = 8$ ✔
4. $x + w = 14 \quad \Rightarrow \quad 6 + 8 = 14$ ✔
5. $z + x = 13 \quad \Rightarrow \quad 7 + 6 = 13$ ✔

Thus, the solution is correct.

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Final Answer:

The code is 6, 0, 7, 8.

Do you want me to explain any step in more detail?

Here are 5 related questions you might find interesting:

1. How do you solve systems of equations with substitution?
2. What is the elimination method for solving systems of equations?
3. How can you represent real-world problems with systems of equations?
4. How do you verify solutions to systems of linear equations?
5. What are some applications of linear equations in daily life?

Tip: Always double-check your solution by substituting values back into the original equations!