To solve the problem, we will determine the smallest possible value for $b$ under the given constraints.

Problem Breakdown:

• $a, b, c$ are single-digit numbers ($1 \leq a, b, c \leq 9$).
• The equation is given as: $3a + 2b + c = 42$

We aim to minimize $b$. To achieve this:

1. Maximize $a$, since $a$ has the largest coefficient ($3a$).
2. Minimize $c$, since $c$ has the smallest coefficient ($c$).

Step 1: Maximize $a$

The maximum value of $a$ is $9$. Substituting $a = 9$: $3(9) + 2b + c = 42$ $27 + 2b + c = 42$ $2b + c = 15$

Step 2: Minimize $b$

To minimize $b$, $c$ should be maximized. The maximum value of $c$ is $9$. Substituting $c = 9$: $2b + 9 = 15$ $2b = 6 \implies b = 3$

Thus, the smallest value of $b$ is 3.

Verification:

Substitute $a = 9$, $b = 3$, and $c = 9$ into the original equation: $3(9) + 2(3) + 9 = 27 + 6 + 9 = 42$ The solution satisfies the equation.

Let me know if you'd like further details or additional questions.

Related Questions:

1. How would the solution change if the maximum $c$ were limited to 8?
2. What happens if $a$ is restricted to a value smaller than 9?
3. Can $b$ take multiple values if $a, b, c$ are different?
4. How does this type of problem relate to optimization techniques in algebra?
5. What would be the outcome if $a, b, c$ were allowed to include 0?

Tip:

Always evaluate the constraints systematically when minimizing or maximizing values in linear equations.