Use the substitution method to find all solutions of the system \( \left\{\begin{array}{l}y=x+1 \\ x y=6\end{array}\right. \) The solutions of the system are: \( x_{1}=\square, y_{1}=\square \) \( x_{2}=\square \) Question Help: \( \square \) Video , Message instructor Calculator

To solve the given system using the substitution method, we start with the first equation \( y = x + 1 \). We can substitute this expression for \( y \) into the second equation \( xy = 6 \). 1. Substituting \( y \): \[ x(x + 1) = 6 \] This expands to: \[ x^2 + x - 6 = 0 \] 2. Now we can factor the quadratic equation: \[ (x - 2)(x + 3) = 0 \] So, the solutions for \( x \) are: \[ x_1 = 2, \quad x_2 = -3 \] 3. Now substituting these \( x \) values back into the equation \( y = x + 1 \): - For \( x_1 = 2 \): \[ y_1 = 2 + 1 = 3 \] - For \( x_2 = -3 \): \[ y_2 = -3 + 1 = -2 \] Thus, the solutions of the system are: \[ x_1 = 2, y_1 = 3 \] \[ x_2 = -3, y_2 = -2 \]