Consider the system of equations. \( 3 x+5 y=2 \) \( y=\frac{2}{3} x-\frac{14}{3} \) What is the solution to the given system of equations? \( (1,-2) \) \( (0,-4) \) \( (7,0) \)

Let's solve the given system of equations to find the correct solution. **Given Equations:** 1. \( 3x + 5y = 2 \) 2. \( y = \frac{2}{3}x - \frac{14}{3} \) **Step 1: Substitute Equation 2 into Equation 1** Replace \( y \) in the first equation with the expression from the second equation: \[ 3x + 5\left(\frac{2}{3}x - \frac{14}{3}\right) = 2 \] **Step 2: Simplify the Equation** \[ 3x + \frac{10}{3}x - \frac{70}{3} = 2 \] Multiply every term by 3 to eliminate the denominators: \[ 9x + 10x - 70 = 6 \] Combine like terms: \[ 19x - 70 = 6 \] Add 70 to both sides: \[ 19x = 76 \] Divide by 19: \[ x = 4 \] **Step 3: Find \( y \)** Substitute \( x = 4 \) back into Equation 2: \[ y = \frac{2}{3}(4) - \frac{14}{3} = \frac{8}{3} - \frac{14}{3} = -\frac{6}{3} = -2 \] **Solution:** \[ (x, y) = (4, -2) \] **Verification of Given Options:** 1. **Option (1, -2):** - Substitute into Equation 1: \( 3(1) + 5(-2) = 3 - 10 = -7 \neq 2 \) → **Does not satisfy.** 2. **Option (0, -4):** - Substitute into Equation 1: \( 3(0) + 5(-4) = 0 - 20 = -20 \neq 2 \) → **Does not satisfy.** 3. **Option (7, 0):** - Substitute into Equation 1: \( 3(7) + 5(0) = 21 + 0 = 21 \neq 2 \) → **Does not satisfy.** None of the provided options \((1, -2)\), \((0, -4)\), or \((7, 0)\) satisfy both equations in the system. **Correct Solution:** \[ (4, -2) \]