Answer :
## Let's solve each equation step-by-step:
### 1. [tex]\( -2 \cdot 6^8 + 8x_1 + 1 = 0 \)[/tex]
To find [tex]\( x_1 \)[/tex]:
[tex]\[ -2 \cdot 6^8 + 8x_1 + 1 = 0 \][/tex]
[tex]\[ 8x_1 = 2 \cdot 6^8 - 1 \][/tex]
[tex]\[ x_1 = \frac{2 \cdot 6^8 - 1}{8} \][/tex]
Given:
[tex]\[ x_1 = 419903.875 \][/tex]
### 2. [tex]\( 3 - 2x_2 - 9 = 0 \)[/tex]
To find [tex]\( x_2 \)[/tex]:
[tex]\[ 3 - 2x_2 - 9 = 0 \][/tex]
[tex]\[ -2x_2 = 6 \][/tex]
[tex]\[ x_2 = -3 \][/tex]
### 3. [tex]\( -3 - \frac{x_3}{10} + 1 = 0 \)[/tex]
To find [tex]\( x_3 \)[/tex]:
[tex]\[ -3 - \frac{x_3}{10} + 1 = 0 \][/tex]
[tex]\[ -2 - \frac{x_3}{10} = 0 \][/tex]
[tex]\[ \frac{x_3}{10} = -2 \][/tex]
[tex]\[ x_3 = -20 \][/tex]
### 4. [tex]\( 18 + 2x_4 - 2 = 0 \)[/tex]
To find [tex]\( x_4 \)[/tex]:
[tex]\[ 18 + 2x_4 - 2 = 0 \][/tex]
[tex]\[ 16 + 2x_4 = 0 \][/tex]
[tex]\[ 2x_4 = -16 \][/tex]
[tex]\[ x_4 = -8 \][/tex]
### 5. [tex]\( 3x_5 + 9 - 45 = 0 \)[/tex]
To find [tex]\( x_5 \)[/tex]:
[tex]\[ 3x_5 + 9 - 45 = 0 \][/tex]
[tex]\[ 3x_5 - 36 = 0 \][/tex]
[tex]\[ 3x_5 = 36 \][/tex]
[tex]\[ x_5 = 12 \][/tex]
### 6. [tex]\( 3 - 6 - \frac{x_6}{5} = 0 \)[/tex]
To find [tex]\( x_6 \)[/tex]:
[tex]\[ 3 - 6 - \frac{x_6}{5} = 0 \][/tex]
[tex]\[ -3 - \frac{x_6}{5} = 0 \][/tex]
[tex]\[ \frac{x_6}{5} = -3 \][/tex]
[tex]\[ x_6 = -15 \][/tex]
### 7. [tex]\( 5 \cdot \frac{x_7}{4} - 2 = 0 \)[/tex]
To find [tex]\( x_7 \)[/tex]:
[tex]\[ 5 \cdot \frac{x_7}{4} - 2 = 0 \][/tex]
[tex]\[ \frac{5x_7}{4} = 2 \][/tex]
[tex]\[ 5x_7 = 8 \][/tex]
[tex]\[ x_7 = \frac{8}{5} \][/tex]
### 8. [tex]\( -6x_8 + 12 + 6 = 0 \)[/tex]
To find [tex]\( x_8 \)[/tex]:
[tex]\[ -6x_8 + 12 + 6 = 0 \][/tex]
[tex]\[ -6x_8 + 18 = 0 \][/tex]
[tex]\[ -6x_8 = -18 \][/tex]
[tex]\[ x_8 = 3 \][/tex]
### 9. [tex]\( -2 + \frac{x_9}{2} - 2 = 0 \)[/tex]
To find [tex]\( x_9 \)[/tex]:
[tex]\[ -2 + \frac{x_9}{2} - 2 = 0 \][/tex]
[tex]\[ -4 + \frac{x_9}{2} = 0 \][/tex]
[tex]\[ \frac{x_9}{2} = 4 \][/tex]
[tex]\[ x_9 = 8 \][/tex]
### 10. [tex]\( 23 + 32 + 5x_{10} = 0 \)[/tex]
To find [tex]\( x_{10} \)[/tex]:
[tex]\[ 23 + 32 + 5x_{10} = 0 \][/tex]
[tex]\[ 55 + 5x_{10} = 0 \][/tex]
[tex]\[ 5x_{10} = -55 \][/tex]
[tex]\[ x_{10} = -11 \][/tex]
### 11. [tex]\( \frac{x_{11}}{6} + 6 = 0 \)[/tex]
To find [tex]\( x_{11} \)[/tex]:
[tex]\[ \frac{x_{11}}{6} + 6 = 0 \][/tex]
[tex]\[ \frac{x_{11}}{6} = -6 \][/tex]
[tex]\[ x_{11} = -36 \][/tex]
Thus, the solutions for the given equations are:
[tex]\[ \begin{aligned} x_1 & = 419903.875 \\ x_2 & = -3 \\ x_3 & = -20 \\ x_4 & = -8 \\ x_5 & = 12 \\ x_6 & = -15 \\ x_7 & = \frac{8}{5} \\ x_8 & = 3 \\ x_9 & = 8 \\ x_{10} & = -11 \\ x_{11} & = -36 \\ \end{aligned} \][/tex]
### 1. [tex]\( -2 \cdot 6^8 + 8x_1 + 1 = 0 \)[/tex]
To find [tex]\( x_1 \)[/tex]:
[tex]\[ -2 \cdot 6^8 + 8x_1 + 1 = 0 \][/tex]
[tex]\[ 8x_1 = 2 \cdot 6^8 - 1 \][/tex]
[tex]\[ x_1 = \frac{2 \cdot 6^8 - 1}{8} \][/tex]
Given:
[tex]\[ x_1 = 419903.875 \][/tex]
### 2. [tex]\( 3 - 2x_2 - 9 = 0 \)[/tex]
To find [tex]\( x_2 \)[/tex]:
[tex]\[ 3 - 2x_2 - 9 = 0 \][/tex]
[tex]\[ -2x_2 = 6 \][/tex]
[tex]\[ x_2 = -3 \][/tex]
### 3. [tex]\( -3 - \frac{x_3}{10} + 1 = 0 \)[/tex]
To find [tex]\( x_3 \)[/tex]:
[tex]\[ -3 - \frac{x_3}{10} + 1 = 0 \][/tex]
[tex]\[ -2 - \frac{x_3}{10} = 0 \][/tex]
[tex]\[ \frac{x_3}{10} = -2 \][/tex]
[tex]\[ x_3 = -20 \][/tex]
### 4. [tex]\( 18 + 2x_4 - 2 = 0 \)[/tex]
To find [tex]\( x_4 \)[/tex]:
[tex]\[ 18 + 2x_4 - 2 = 0 \][/tex]
[tex]\[ 16 + 2x_4 = 0 \][/tex]
[tex]\[ 2x_4 = -16 \][/tex]
[tex]\[ x_4 = -8 \][/tex]
### 5. [tex]\( 3x_5 + 9 - 45 = 0 \)[/tex]
To find [tex]\( x_5 \)[/tex]:
[tex]\[ 3x_5 + 9 - 45 = 0 \][/tex]
[tex]\[ 3x_5 - 36 = 0 \][/tex]
[tex]\[ 3x_5 = 36 \][/tex]
[tex]\[ x_5 = 12 \][/tex]
### 6. [tex]\( 3 - 6 - \frac{x_6}{5} = 0 \)[/tex]
To find [tex]\( x_6 \)[/tex]:
[tex]\[ 3 - 6 - \frac{x_6}{5} = 0 \][/tex]
[tex]\[ -3 - \frac{x_6}{5} = 0 \][/tex]
[tex]\[ \frac{x_6}{5} = -3 \][/tex]
[tex]\[ x_6 = -15 \][/tex]
### 7. [tex]\( 5 \cdot \frac{x_7}{4} - 2 = 0 \)[/tex]
To find [tex]\( x_7 \)[/tex]:
[tex]\[ 5 \cdot \frac{x_7}{4} - 2 = 0 \][/tex]
[tex]\[ \frac{5x_7}{4} = 2 \][/tex]
[tex]\[ 5x_7 = 8 \][/tex]
[tex]\[ x_7 = \frac{8}{5} \][/tex]
### 8. [tex]\( -6x_8 + 12 + 6 = 0 \)[/tex]
To find [tex]\( x_8 \)[/tex]:
[tex]\[ -6x_8 + 12 + 6 = 0 \][/tex]
[tex]\[ -6x_8 + 18 = 0 \][/tex]
[tex]\[ -6x_8 = -18 \][/tex]
[tex]\[ x_8 = 3 \][/tex]
### 9. [tex]\( -2 + \frac{x_9}{2} - 2 = 0 \)[/tex]
To find [tex]\( x_9 \)[/tex]:
[tex]\[ -2 + \frac{x_9}{2} - 2 = 0 \][/tex]
[tex]\[ -4 + \frac{x_9}{2} = 0 \][/tex]
[tex]\[ \frac{x_9}{2} = 4 \][/tex]
[tex]\[ x_9 = 8 \][/tex]
### 10. [tex]\( 23 + 32 + 5x_{10} = 0 \)[/tex]
To find [tex]\( x_{10} \)[/tex]:
[tex]\[ 23 + 32 + 5x_{10} = 0 \][/tex]
[tex]\[ 55 + 5x_{10} = 0 \][/tex]
[tex]\[ 5x_{10} = -55 \][/tex]
[tex]\[ x_{10} = -11 \][/tex]
### 11. [tex]\( \frac{x_{11}}{6} + 6 = 0 \)[/tex]
To find [tex]\( x_{11} \)[/tex]:
[tex]\[ \frac{x_{11}}{6} + 6 = 0 \][/tex]
[tex]\[ \frac{x_{11}}{6} = -6 \][/tex]
[tex]\[ x_{11} = -36 \][/tex]
Thus, the solutions for the given equations are:
[tex]\[ \begin{aligned} x_1 & = 419903.875 \\ x_2 & = -3 \\ x_3 & = -20 \\ x_4 & = -8 \\ x_5 & = 12 \\ x_6 & = -15 \\ x_7 & = \frac{8}{5} \\ x_8 & = 3 \\ x_9 & = 8 \\ x_{10} & = -11 \\ x_{11} & = -36 \\ \end{aligned} \][/tex]
