What is the solution to the system of equations? \( \begin{array}{l}4 x+2 y=-1 \\ 3 x+4 y=3 \\ \text { A. }\left(1, \frac{2}{3}\right) \\ \text { B. }\left(-1, \frac{2}{3}\right) \\ \text { C. }\left(1, \frac{3}{2}\right) \\ \text { D. }\left(-1, \frac{3}{2}\right)\end{array} \)

To solve the system of equations: 1. First, manipulate the first equation, \( 4x + 2y = -1 \), to express \( y \) in terms of \( x \): \[ 2y = -1 - 4x \implies y = -\frac{1 + 4x}{2} \] 2. Substitute this expression for \( y \) into the second equation, \( 3x + 4y = 3 \): \[ 3x + 4\left(-\frac{1 + 4x}{2}\right) = 3 \] Simplifying that gives: \[ 3x - 2(1 + 4x) = 3 \implies 3x - 2 - 8x = 3 \implies -5x - 2 = 3 \] Thus, \[ -5x = 5 \implies x = -1 \] 3. Substitute \( x = -1 \) back into the equation for \( y \): \[ y = -\frac{1 + 4(-1)}{2} = -\frac{1 - 4}{2} = -\frac{-3}{2} = \frac{3}{2} \] So the solution to the system of equations is \( \left(-1, \frac{3}{2}\right) \) which corresponds to option D. Now, for a bit more engagement: Remember that the method you used is called substitution, and it's super handy! It's like being a detective, piecing together clues from different equations to solve the mystery of the unknown variables. Also, when diving into solving systems of equations, checking your work is a golden rule. After finding that \( x = -1 \) and \( y = \frac{3}{2} \), plug these values back into the original equations to confirm they hold true; this is your verification badge of honor! Happy solving!