Question

Linear

Algebra
Solve the system [ left{begin{array}{rr} x_{1}+x_{2}-4 x_{3}= & -2 6 x_{1}+7 x_{2}+2 x_{3}=9 \end{array}right. ] [ left[begin{array}{l} x_{1} x_{2} x_{3} end{array}right]=[]+s

Answer 1

To solve the system of equations, we can use the method of Gaussian elimination. The solution can be expressed in terms of a parameter.

Step-by-step explanation:

To solve the system of equations, we can use the method of Gaussian elimination or matrix algebra. Let's use Gaussian elimination:

1. Write the system of equations in augmented matrix form:

$$\begin{bmatrix}1 & 1 & -4 & -2 \\ 6 & 7 & 2 & 9 \end{bmatrix}$$

1. Create zeros below the pivot element:

$$\begin{bmatrix}1 & 1 & -4 & -2 \\ 0 & 1 & 26 & 21 \end{bmatrix}$$

1. Eliminate the coefficients above and below the pivot element:

$$\begin{bmatrix}1 & 0 & 70 & 59 \\ 0 & 1 & 26 & 21 \end{bmatrix}$$

1. Write the system of equations in row echelon form:

$$\begin{aligned} x_1 + 70x_3 &= 59 \\ x_2 + 26x_3 &= 21 \end{aligned}$$

1. Now, the system of equations is solved. We can express the solution as:

$$\begin{aligned} x_1 &= 59 - 70x_3 \\ x_2 &= 21 - 26x_3 \\ x_3 &= \text{s (parameter)} \end{aligned}$$