Question: Write a system of equations having \(\{(-4, 2)\}\) as a solution set. Choose the correct…

Write a system of equations having \(\{(-4, 2)\}\) as a solution set.

Choose the correct answer below.

A. \(\left\{ \begin{array}{l} -3x + y = 14 \\ 4x + y = -14 \end{array} \right.\)

B. \(\left\{ \begin{array}{l} 4x + y = 14 \\ 3x + y = -14 \end{array} \right.\)

C. \(\left\{ \begin{array}{l} 4x + y = -14 \\ 3x + y = 14 \end{array} \right.\)

D. \(\left\{ \begin{array}{l} -3x + y = -14 \\ 4x + y = 14 \end{array} \right.\)

Solution

To solve this problem, we need to find which system of equations has the solution \((-4, 2)\). First, we’ll substitute \(x = -4\) and \(y = 2\) into each given system of equations to verify which one satisfies both equations. Option A: \[ \begin{cases} -3x + y = 14 & \\ 4x + y = -14 & \end{cases} \] Substitute \(x = -4\) and \(y = 2\): 1. \(-3(-4) + 2 = 12 + 2 = 14\) (True) 2. \(4(-4) + 2 = -16 + 2 = -14\) (True) Both equations are satisfied, so Option A is correct. Option B: \[ \begin{cases} 4x + y = 14 & \\ 3x + y = -14 & \end{cases} \] Substitute \(x = -4\) and \(y = 2\): 1. \(4(-4) + 2 = -16 + 2 = -14\) (False) 2. \(3(-4) + 2 = -12 + 2 = -10\) (False) Not satisfied. Option C: \[ \begin{cases} 4x + y = -14 & \\ 3x + y = 14 & \end{cases} \] Substitute \(x = -4\) and \(y = 2\): 1. \(4(-4) + 2 = -16 + 2 = -14\) (True) 2. \(3(-4) + 2 = -12 + 2 = -10\) (False) Not satisfied. Option D: \[ \begin{cases} -3x + y = -14 & \\ 4x + y = 14 & \end{cases} \] Substitute \(x = -4\) and \(y = 2\): 1. \(-3(-4) + 2 = 12 + 2 = 14\) (False) 2. \(4(-4) + 2 = -16 + 2 = -14\) (False) Not satisfied. The correct answer is Option A.