If a + b = 5 and 3a + 2b = 20, what is the value of the expression 3a + b?

Introduction / Context:
This question is a simple example of solving simultaneous linear equations and then evaluating a particular expression in terms of the variables. Rather than asking for the individual values of a and b, the exam tests whether you can use the given equations efficiently to find 3a + b. Such questions are routine in algebra sections and help build skill in manipulating equations smartly.

Given Data / Assumptions:

• a + b = 5.
• 3a + 2b = 20.
• We must find the value of 3a + b.
• We assume a and b are real numbers.

Concept / Approach:
There are two main approaches. One is to solve for a and b explicitly by using substitution or elimination, and then compute 3a + b. The other is to manipulate the equations directly to get 3a + b without fully solving for both variables. In either approach, the calculations are straightforward. Here we will solve for a and b explicitly to keep the logic transparent and then evaluate the required expression.

Step-by-Step Solution:
From the first equation, a + b = 5. Express a in terms of b: a = 5 - b. Substitute this into the second equation 3a + 2b = 20. 3(5 - b) + 2b = 20. Expand: 15 - 3b + 2b = 20. Combine like terms: 15 - b = 20. Rearrange to find b: -b = 20 - 15 = 5, so b = -5. Now substitute b = -5 back into a = 5 - b. a = 5 - (-5) = 10. So, a = 10 and b = -5. Now compute the required expression 3a + b. 3a + b = 3 * 10 + (-5) = 30 - 5 = 25.

Verification / Alternative check:
We can quickly verify that a = 10 and b = -5 satisfy both original equations. First equation: a + b = 10 + (-5) = 5, which matches. Second equation: 3a + 2b = 3 * 10 + 2 * (-5) = 30 - 10 = 20, which matches as well. Since the pair (10, -5) satisfies both equations, the computed value 3a + b = 25 is consistent and correct.

Why Other Options Are Wrong:
15 and 20: These might result from partial substitution errors or from using only one equation without checking the second. For example, someone might incorrectly assume b is positive or miscalculate 3a + b after solving.
30 and 10: These could arise from using a + b directly, such as mistakenly treating 3a + b as 3(a + b) or mixing coefficients, both of which give wrong results.

Common Pitfalls:
Common errors include incorrect substitution, such as writing a = 5 + b instead of a = 5 - b, or mismanaging signs during simplification. Some learners rush and compute 3(a + b) instead of 3a + b, which gives 3 * 5 = 15 and leads to an incorrect option. Carefully distinguishing between expressions and using clean algebraic steps prevents these mistakes.

Final Answer:
The value of the expression 3a + b is 25.