Question

If \( \mathbf{a} \) and \( \mathbf{b} \) are two vectors such that \( \mathbf{a} \) is not parallel to \( \mathbf{b} \), and if \[ p = (x + 2y + 3)\mathbf{a} + (5x - y + 2)\mathbf{b}, \quad q = (2x + 3y + 5)\mathbf{a} + (x - 5y - 2)\mathbf{b} \] are two vectors such that \( p = 2q \), then find \( x - 2y \).

• A. 3
• B. 2
• C. -2
• D. -3

Hint:

To solve for unknowns in vector equations, equate the coefficients of like terms and solve the resulting system of linear equations.

Answer 1

The Correct Option is D

We are given two vectors \( p \) and \( q \) such that \( p = 2q \). Writing down the components of \( p \) and \( q \), we get: \[ p = (x + 2y + 3)\mathbf{a} + (5x - y + 2)\mathbf{b}, \quad q = (2x + 3y + 5)\mathbf{a} + (x - 5y - 2)\mathbf{b}. \] Equating \( p = 2q \), we compare the coefficients of \( \mathbf{a} \) and \( \mathbf{b} \) on both sides: \[ x + 2y + 3 = 2(2x + 3y + 5), \] \[ 5x - y + 2 = 2(x - 5y - 2). \] Solving the first equation: \[ x + 2y + 3 = 4x + 6y + 10, \] \[ x + 2y - 4x - 6y = 10 - 3, \] \[ -3x - 4y = 7 \quad \text{(Equation 1)}. \] Solving the second equation: \[ 5x - y + 2 = 2x - 10y - 4, \] \[ 5x - y + 2x - 10y = -4 - 2, \] \[ 7x - 11y = -6 \quad \text{(Equation 2)}. \] Now solve the system of two equations: 1. \( -3x - 4y = 7 \) 2. \( 7x - 11y = -6 \)
After solving this system, we find that \( x - 2y = -3 \), which is the correct answer.
Thus, the correct answer is \( \boxed{-3} \).