Question

5y-4z=3 x+y-z=6 2x-y+3z=-1 решение систем уравнений по формуле Крамера. Как можно быстрее

Answer 1

$\left[\begin{array}{ccc}0&5&-4\\1&1&-1\\2&-1&3\end{array}\right]$$\left[\begin{array}{ccc}3\\6\\-1\end{array}\right]$

$\Delta \left[\begin{array}{ccc}0&5&-4\\1&1&-1\\2&-1&3\end{array}\right]=0*(3-1)-5(3+2)+(-4)(-1-2)=-13$

$\Delta_1 \left[\begin{array}{ccc}3&5&-4\\6&1&-1\\-1&-1&3\end{array}\right]=3(3-1)-5(18-1)+(-4)(-6+1)=-59$

$\Delta_2 \left[\begin{array}{ccc}0&3&-4\\1&6&-1\\2&-1&3\end{array}\right]=0-3(3+2)+(-4)(-1-12)=-15+52=37$

$\Delta_3 \left[\begin{array}{ccc}0&5&3\\1&1&6\\2&-1&-1\end{array}\right]=0-5(-1-12)+3(-1-2)=65-9=56$

$x_1= \frac{\Delta_1}{\Delta}= \frac{-59}{-13}= \frac{59}{13}$

$x_2= \frac{\Delta_2}{\Delta}=- \frac{37}{13}$

$x_3= \frac{\Delta_3}{\Delta}=- \frac{56}{13}$

Answer 2

Solving the System of Equations using Cramer's Rule

To solve the system of equations using Cramer's Rule, we can use the following approach:

1. Identify the Coefficients and Constants: - The given system of equations is: - 5y - 4z = 3 - x + y - z = 6 - 2x - y + 3z = -1

2. Calculate the Determinant (D): - The determinant (D) of the coefficient matrix is calculated as follows: - D = |5 -4 0| = 5(1*3 - (-1)*(-4)) - (-4*(1*(-1) - 3*0)) = 5(7) - (-4) = 35 + 4 = 39.

3. Calculate the Determinant (Dx): - The determinant (Dx) for the x-variable is calculated by replacing the x-coefficients with the constants and finding the determinant: - Dx = |3 -4 0| = 3(1*3 - (-1)*(-4)) - (-4*(1*(-1) - 3*0)) = 3(7) - (-4) = 21 + 4 = 25.

4. Calculate the Determinant (Dy): - The determinant (Dy) for the y-variable is calculated by replacing the y-coefficients with the constants and finding the determinant: - Dy = |5 3 0| = 5(1*6 - (-1)*3) - 3(1*6 - (-1)*3) = 5(9) - 3(9) = 45 - 27 = 18.

5. Calculate the Determinant (Dz): - The determinant (Dz) for the z-variable is calculated by replacing the z-coefficients with the constants and finding the determinant: - Dz = |5 -4 3| = 5(1*(-1) - 6*(-4)) - (-4*(1*(-1) - 3*6)) = 5(25) - (-4*(-25)) = 125 + 100 = 225.

6. Calculate the Values of x, y, and z: - The values of x, y, and z can be calculated using the following formulas: - x = Dx / D = 25 / 39 - y = Dy / D = 18 / 39 - z = Dz / D = 225 / 39

The solutions for the given system of equations using Cramer's Rule are: - x = 25 / 39 - y = 18 / 39 - z = 225 / 39