Solve the system of equations: \begin{cases} y^2 - xy + x = 2 \\ 5y + x = 12 \end{cases}

Let's solve this system of equations!

Step-by-step solution:

1. Step 1: Express x in terms of y from the second equation: \[ x = 12 - 5y \]
2. Step 2: Substitute this expression for x into the first equation: \[ y^2 - y(12 - 5y) + (12 - 5y) = 2 \]
3. Step 3: Simplify the equation: \[ y^2 - 12y + 5y^2 + 12 - 5y = 2 \] \[ 6y^2 - 17y + 10 = 0 \]
4. Step 4: Solve the quadratic equation for y. We can use the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where a = 6, b = -17, and c = 10. \[ y = \frac{17 \pm \sqrt{(-17)^2 - 4(6)(10)}}{2(6)} \] \[ y = \frac{17 \pm \sqrt{289 - 240}}{12} \] \[ y = \frac{17 \pm \sqrt{49}}{12} \] \[ y = \frac{17 \pm 7}{12} \] So, we have two possible values for y: \[ y_1 = \frac{17 + 7}{12} = \frac{24}{12} = 2 \] \[ y_2 = \frac{17 - 7}{12} = \frac{10}{12} = \frac{5}{6} \]
5. Step 5: Find the corresponding values for x using the equation x = 12 - 5y: For y_1 = 2: \[ x_1 = 12 - 5(2) = 12 - 10 = 2 \] For y_2 = \frac{5}{6}: \[ x_2 = 12 - 5(\frac{5}{6}) = 12 - \frac{25}{6} = \frac{72 - 25}{6} = \frac{47}{6} \]

Answer: The solutions are (x, y) = (2, 2) and (x, y) = (\(\frac{47}{6}\), \(\frac{5}{6}\)).