We are given a system of six linear equations with six variables, $x_1, x_2, x_3, x_4, x_5, x_6$. The system is: $x_2 + x_3 + x_4 + x_5 + x_6 = 3$ $x_1 + x_3 + x_4 + x_5 + x_6 = 5$ $x_1 + x_2 + x_4 + x_5 + x_6 = 4$ $x_1 + x_2 + x_3 + x_5 + x_6 = 7$ $x_1 + x_2 + x_3 + x_4 + x_6 = 11$ $x_1 + x_2 + x_3 + x_4 + x_5 = 10$ We need to find the values of the six variables.

1. Problem Description

We are given a system of six linear equations with six variables, $x_1, x_2, x_3, x_4, x_5, x_6$. The system is:

$x_2 + x_3 + x_4 + x_5 + x_6 = 3$

$x_1 + x_3 + x_4 + x_5 + x_6 = 5$

$x_1 + x_2 + x_4 + x_5 + x_6 = 4$

$x_1 + x_2 + x_3 + x_5 + x_6 = 7$

$x_1 + x_2 + x_3 + x_4 + x_6 = 11$

$x_1 + x_2 + x_3 + x_4 + x_5 = 10$

We need to find the values of the six variables.

2. Solution Steps

Let's denote the sum of all variables by $S = x_1 + x_2 + x_3 + x_4 + x_5 + x_6$.

From the given equations, we can express each equation in terms of $S$ and one variable:

$S - x_1 = 3$

$S - x_2 = 5$

$S - x_3 = 4$

$S - x_4 = 7$

$S - x_5 = 11$

$S - x_6 = 10$

Now we can express each variable in terms of $S$:

$x_1 = S - 3$

$x_2 = S - 5$

$x_3 = S - 4$

$x_4 = S - 7$

$x_5 = S - 11$

$x_6 = S - 10$

Substitute these expressions into the equation for $S$:

$S = (S - 3) + (S - 5) + (S - 4) + (S - 7) + (S - 11) + (S - 10)$

$S = 6S - (3 + 5 + 4 + 7 + 11 + 10)$

$S = 6S - 40$

$5S = 40$

$S = 8$

Now we can find the values of the variables:

$x_1 = S - 3 = 8 - 3 = 5$

$x_2 = S - 5 = 8 - 5 = 3$

$x_3 = S - 4 = 8 - 4 = 4$

$x_4 = S - 7 = 8 - 7 = 1$

$x_5 = S - 11 = 8 - 11 = -3$

$x_6 = S - 10 = 8 - 10 = -2$

3. Final Answer

$x_1 = 5$

$x_2 = 3$

$x_3 = 4$

$x_4 = 1$

$x_5 = -3$

$x_6 = -2$