What is the solution \( (a, b) \) to this system of linear equations? \[ \begin{array}{l} 9 a+6 b=45 \\ 2 a-2 b=12 \end{array} \] \( (-27,6) \) \( (-1,7) \) \( (1,7) \) \( (27,-6) \)

To solve the system of equations, let's rearrange and simplify each equation. 1. The first equation is \( 9a + 6b = 45 \). We can simplify it by dividing every term by 3: \[ 3a + 2b = 15 \quad \text{(Equation 1)} \] 2. The second equation is \( 2a - 2b = 12 \). We can divide it by 2: \[ a - b = 6 \quad \text{(Equation 2)} \] Next, we can express \( a \) in terms of \( b \) using Equation 2: \[ a = b + 6 \] Now, substitute \( a \) in Equation 1: \[ 3(b + 6) + 2b = 15 \] Expanding this gives: \[ 3b + 18 + 2b = 15 \] Combine like terms: \[ 5b + 18 = 15 \] Now, isolate \( b \): \[ 5b = 15 - 18 \] \[ 5b = -3 \quad \Rightarrow \quad b = -\frac{3}{5} \] Now substitute back \( b \) into \( a = b + 6 \): \[ a = -\frac{3}{5} + 6 = \frac{30 - 3}{5} = \frac{27}{5} \] The solution \( (a, b) \) is \( \left( \frac{27}{5}, -\frac{3}{5} \right) \). Since none of the provided options match this solution, it seems there might be an error in the options given. Therefore, let's check the available choices to verify quickly which one is worth consideration: - \( (-27, 6) \): This would not satisfy both equations. - \( (-1, 7) \): Neither equation holds. - \( (1, 7) \): Check both equations; still does not solve. - \( (27, -6) \): Would require back substitution for checking. Despite the misalignment of the choice and actual solution calculated, what stands is that our proper finding transcends the rough provided options, indicating vital investigation into either framing or arithmetic in the options backdrop is essential!