We are tasked with solving a system of equations involving four digits, where:

1. $2^{\text{nd}} \text{ digit} + 1^{\text{st}} \text{ digit} = 13$
2. $1^{\text{st}} \text{ digit} + 4^{\text{th}} \text{ digit} = 10$
3. $3^{\text{rd}} \text{ digit} + 2^{\text{nd}} \text{ digit} = 12$
4. $3^{\text{rd}} \text{ digit} + 1^{\text{st}} \text{ digit} = 17$
5. $2^{\text{nd}} \text{ digit} + 4^{\text{th}} \text{ digit} = 5$

Let's solve this step by step.

Step 1: Define Variables

Let:

• $x$ = 1st digit
• $y$ = 2nd digit
• $z$ = 3rd digit
• $w$ = 4th digit

The equations become:

1. $y + x = 13$
2. $x + w = 10$
3. $z + y = 12$
4. $z + x = 17$
5. $y + w = 5$

Step 2: Solve the System of Equations

From Equation (1):

$y + x = 13 \implies y = 13 - x$

Substitute $y = 13 - x$ into Equation (5):

$(13 - x) + w = 5$ $w = 5 - 13 + x \implies w = x - 8$

Now substitute $y = 13 - x$ into Equation (3):

$z + (13 - x) = 12$ $z = 12 - 13 + x \implies z = x - 1$

Substitute $z = x - 1$ into Equation (4):

$(x - 1) + x = 17$ $2x - 1 = 17$ $2x = 18 \implies x = 9$

Step 3: Solve for Other Variables

Now that $x = 9$:

1. From $y = 13 - x$: $y = 13 - 9 = 4$
2. From $w = x - 8$: $w = 9 - 8 = 1$
3. From $z = x - 1$: $z = 9 - 1 = 8$

Step 4: Verify the Solution

• $y + x = 4 + 9 = 13$ ✔
• $x + w = 9 + 1 = 10$ ✔
• $z + y = 8 + 4 = 12$ ✔
• $z + x = 8 + 9 = 17$ ✔
• $y + w = 4 + 1 = 5$ ✔

Final Solution:

The digits are:

• 1st digit ($x$) = 9
• 2nd digit ($y$) = 4
• 3rd digit ($z$) = 8
• 4th digit ($w$) = 1

Would you like me to explain any step in further detail? Let me know!

Here are 5 related questions to deepen your understanding:

1. How can you solve systems of equations using substitution and elimination?
2. What are the different methods to solve a system of linear equations?
3. Can you solve systems of equations graphically?
4. How do you check if a solution to a system of equations is correct?
5. What are real-life applications of solving systems of equations?

Tip: Always verify your solution by substituting the values back into the original equations!