Question

What is the value of \(2x+3y\)? I) \(x+y=2\).
II) \(3x-2y=1\).

• A. Statement I alone is sufficient to answer the question
• B. Statement II alone is sufficient to answer the question
• C. Both the statements I and II are sufficient to answer the question but neither statement alone is not sufficient
• D. Both the statements I and II together are not sufficient to answer the question and additional data is required

Hint:

Two independent linear equations are generally sufficient to determine two unknowns.

Answer 1

The Correct Option is C

Concept:
To find the value of \[ 2x+3y \] we need enough information to determine \(x\) and \(y\), or directly determine the required expression.

Step 1: Check Statement I. Statement I gives \[ x+y=2 \] This is one equation in two unknowns. There are infinitely many pairs \((x,y)\), so \(2x+3y\) cannot be uniquely determined. Thus Statement I alone is not sufficient.

Step 2: Check Statement II. Statement II gives \[ 3x-2y=1 \] This is also one equation in two unknowns. So Statement II alone is not sufficient.

Step 3: Combine both statements. We solve \[ x+y=2 \] \[ 3x-2y=1 \] From the first equation, \[ x=2-y \] Substitute in the second equation: \[ 3(2-y)-2y=1 \] \[ 6-3y-2y=1 \] \[ 6-5y=1 \] \[ 5y=5 \] \[ y=1 \] Then, \[ x+y=2 \] \[ x+1=2 \] \[ x=1 \] Now, \[ 2x+3y=2(1)+3(1)=5 \] Thus both statements together are sufficient.

Step 4: Final answer. \[ \boxed{\text{Both statements together are sufficient, but neither alone is sufficient}} \]