Questions: Используя метод подстановки, решите систему линейных уравнений: 3x - 6y = 5/2 -6x + 3y = 1 Выразите x из первого уравнения: x = -6x + 3y = 1 и найдите решение системы линейных уравнений. x= , y=

Solution Steps

We start with the system of linear equations: \[ \left\{ \begin{aligned} 3x - 6y & = \frac{5}{2} \\ -6x + 3y & = 1 \end{aligned} \right. \]

We convert the system into an augmented matrix form: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 3 & -6 & \frac{5}{2} \\ -6 & 3 & 1 \\ \end{array} \right] \]

We perform row operations to simplify the augmented matrix. The first step is to make the leading coefficient of the first row equal to 1: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -2 & \frac{5}{6} \\ -6 & 3 & 1 \\ \end{array} \right] \]

Next, we eliminate the first variable from the second row: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -2 & \frac{5}{6} \\ 0 & -9 & 6 \\ \end{array} \right] \]

Now, we simplify the second row: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -2 & \frac{5}{6} \\ 0 & 1 & -\frac{2}{3} \\ \end{array} \right] \]

Finally, we eliminate the second variable from the first row: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & 0 & -\frac{1}{2} \\ 0 & 1 & -\frac{2}{3} \\ \end{array} \right] \]

From the final augmented matrix, we can read the solutions: \[ x = -\frac{1}{2}, \quad y = -\frac{2}{3} \]

Final Answer