Question

Given the system of equations:
\(x + y = 12\)
\(2x - y = 6\)

Quantity A: x
Quantity B: y

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

Hint:

Always look at the structure of the system of equations before starting. If you see variables with coefficients that are the same or opposites, the elimination method will almost always be the fastest way to solve the system.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to solve a system of two linear equations to find the values of x and y, and then compare them.
Step 2: Key Formula or Approach:
The elimination method is very efficient here because the y-coefficients are opposites (+1 and -1). We can add the two equations together to eliminate y.
Equation 1: \(x + y = 12\)
Equation 2: \(2x - y = 6\)
Step 3: Detailed Explanation:
Add Equation 1 and Equation 2 directly: \[ (x + y) + (2x - y) = 12 + 6 \] The y-terms cancel out: \[ 3x = 18 \] Solve for x by dividing by 3: \[ x = 6 \] Now substitute the value of x back into either of the original equations to find y. Using Equation 1 is simpler: \[ 6 + y = 12 \] Subtract 6 from both sides: \[ y = 6 \]
Comparison:
Quantity A: x = 6
Quantity B: y = 6
The two quantities are equal.
Step 4: Final Answer:
The two quantities are equal.