Question

If \( 4x + 2y = 14 \) and \( \frac{5x}{2y} = \frac{5}{3} \), find the value of \( y - x \).

• A. 3
• B. -5
• C. 1
• D. 5

Hint:

Simplify both given equations to the standard form ax + by = c first, then solve the two linear equations together.

Answer 1

The Correct Option is C

Step 1: Simplify the first equation.
Divide \( 4x + 2y = 14 \) throughout by 2.
\[ 2x + y = 7 \quad \text{...(i)} \]

Step 2: Simplify the second equation.
Cross multiply \( \frac{5x}{2y} = \frac{5}{3} \).
\[ 5x \times 3 = 5 \times 2y \Rightarrow 15x = 10y \Rightarrow 3x - 2y = 0 \quad \text{...(ii)} \]

Step 3: Eliminate one variable.
Multiply (i) by 2 so the y terms can cancel with (ii).
\[ 4x + 2y = 14 \]
Add this to (ii):
\[ (4x+2y) + (3x-2y) = 14 + 0 \Rightarrow 7x = 14 \Rightarrow x = 2 \]

Step 4: Find y using equation (i).
\[ 2(2) + y = 7 \Rightarrow 4 + y = 7 \Rightarrow y = 3 \]

Step 5: Find y - x.
\[ y - x = 3 - 2 = 1 \]

Final Answer:
The value of y - x is 1. \[ \boxed{1} \]