Question

Given the system of equations:
\(3x + 4y = 5\)
\(x - y = 6\)

Quantity A: x
Quantity B: y

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

Hint:

When comparing a positive and a negative number, the positive number is always greater. Once you found that x was positive and y was negative, you didn't need to worry about their exact fractional values to make the comparison.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We are given a system of two linear equations and asked to compare the values of the variables x and y. We need to solve the system to find the unique values for x and y.
Step 2: Key Formula or Approach:
We can use either the substitution or elimination method. The substitution method seems straightforward here.
Equation 1: \(3x + 4y = 5\)
Equation 2: \(x - y = 6\)
Step 3: Detailed Explanation:
From Equation 2, we can easily isolate x: \[ x = y + 6 \] Now, substitute this expression for x into Equation 1: \[ 3(y + 6) + 4y = 5 \] Distribute the 3: \[ 3y + 18 + 4y = 5 \] Combine the y-terms: \[ 7y + 18 = 5 \] Subtract 18 from both sides: \[ 7y = 5 - 18 \] \[ 7y = -13 \] \[ y = -\frac{13}{7} \] Now that we have the value of y, substitute it back into the expression for x: \[ x = y + 6 = -\frac{13}{7} + 6 = -\frac{13}{7} + \frac{42}{7} = \frac{29}{7} \]
Comparison:
Quantity A: \(x = \frac{29}{7}\) (a positive number)
Quantity B: \(y = -\frac{13}{7}\) (a negative number)
Any positive number is greater than any negative number.
Step 4: Final Answer:
Quantity A is greater.