(2) \( 12 x-4 y=28 \) \( 12 x-9 y-33=0 \)

To solve the system of equations, we can first rewrite the second equation: 1. From the equation \( 12x - 9y - 33 = 0 \), we can express \( y \) in terms of \( x \): \[ 9y = 12x - 33 \quad \Rightarrow \quad y = \frac{12x - 33}{9} = \frac{4x - 11}{3} \] 2. Now, substitute this expression for \( y \) into the first equation \( 12x - 4y = 28 \): \[ 12x - 4\left(\frac{4x - 11}{3}\right) = 28 \] Simplifying this will help us find the value of \( x \): \[ 12x - \frac{16x - 44}{3} = 28 \] Multiply through by 3 to eliminate the fraction: \[ 36x - (16x - 44) = 84 \] This leads to: \[ 36x - 16x + 44 = 84 \quad \Rightarrow \quad 20x + 44 = 84 \quad \Rightarrow \quad 20x = 40 \quad \Rightarrow \quad x = 2 \] 3. Substitute \( x = 2 \) back into the equation for \( y \): \[ y = \frac{4(2) - 11}{3} = \frac{8 - 11}{3} = \frac{-3}{3} = -1 \] The solution to the system of equations is \( x = 2 \) and \( y = -1 \). Whether you’re using systems of equations in algebra or tackling complex real-world scenarios, understanding how two or more variables interact can be incredibly fulfilling! From calculating expenses in finance to determining optimal routes in logistics, this skill is as practical as it is academic. While solving these equations, a common mistake is forgetting to line up coefficients correctly when combining or eliminating variables. Always check your multiplication and addition steps to avoid small errors that can lead to incorrect solutions. Happy solving!