If 5x + 3y=4 and 5x-3y= 2, what is the value of (25x$^2$ -9y$^2$)?

Analyze the question: We are given two equations and asked for 25x² − 9y². Notice that this expression is a difference of two squares.

Concept: a² − b² = (a + b)(a − b)

Here, 25x² = (5x)² and 9y² = (3y)², so:

25x² − 9y² = (5x + 3y)(5x − 3y)

Solution:

Substitute the given values directly:

(5x + 3y)(5x − 3y) = 4 × 2 = 8

Check (optional): Adding the equations gives 10x = 6, so x = 0.6. Then 3y = 4 − 3 = 1, so y = 1/3.

• 25x² = 25(0.36) = 9
• 9y² = 9(1/9) = 1
• 9 − 1 = 8 ✓

Answer: D (8)

Tip: Whenever you see a difference of two squares in the target expression and the given equations look like (a + b) and (a − b), don't solve for x and y. Just multiply the two equations. This saves time in the exam.