Question

\(2x + 3y = 29\)
\(3x + 4y = 41\)
Column A: \(x + y\)
Column B: 12

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

Before jumping into a full solution for x and y, look for shortcuts. Sometimes, adding or subtracting the given equations can directly yield the expression you need to evaluate, saving a lot of time.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We are given a system of two linear equations with two variables, x and y. We need to find the value of the expression \(x+y\) and compare it to 12.
Step 2: Key Formula or Approach:
We can solve this system by elimination or substitution. A quicker approach might be to manipulate the equations directly to find the expression \(x+y\).
Step 3: Detailed Explanation:
Let the given equations be:
(1) \(2x + 3y = 29\)
(2) \(3x + 4y = 41\)
Notice that the coefficients of x and y in the second equation are larger than in the first. Let's try subtracting the first equation from the second equation.
\[ (3x + 4y) - (2x + 3y) = 41 - 29 \]
Distribute the negative sign on the left side:
\[ 3x + 4y - 2x - 3y = 12 \]
Group like terms:
\[ (3x - 2x) + (4y - 3y) = 12 \]
\[ x + y = 12 \]
This gives us the value of the expression in Column A directly.
Comparison:
Column A: \(x + y = 12\)
Column B: 12
The two quantities are equal.
Step 4: Final Answer:
The two quantities are equal.