Systems of three equations in three unknowns

Check whether the triple $x = 1$, $y = -1$, $z = 2$ is a solution to the following system. Justify your answer.

$3x + y - z = 0$

$x - 2y + z = 5$

$2x + y + z = 3$

Solve the following system of linear equations, which is already in echelon form, using back-substitution:

$x + 2y - z = 4$

$3y + z = 5$

$2z = 4$

Apply Gaussian elimination to transform into triangular form and solve the following system of equations:

$x - y + z = 2$

$2x + y - z = 1$

$x + 2y + z = 7$

Solve the following system of linear equations containing fractional coefficients. Simplify the equations before applying Gaussian elimination:

$\frac{x}{2} + \frac{y}{3} - z = 1$

$x - \frac{y}{2} + \frac{z}{4} = 2$

$\frac{x}{3} + y - \frac{z}{2} = 3$

Transform the following system into triangular form using Gaussian elimination and state whether it is an Inconsistent System:

$x + y + z = 1$

$2x + y - z = 0$

$3x + 2y = 5$

Apply Gaussian elimination to the following system and check that it is a consistent dependent system by obtaining an equation of the form $0 = 0$:

$x - y + 2z = 3$

$-2x + 2y - 4z = -6$

$3x - 3y + 6z = 9$

Solve the following consistent dependent system and express its infinitely many solutions in terms of a parameter $\lambda$:

$x + 2y - z = 4$

$2x - y + 3z = 3$

$3x + y + 2z = 7$

The sum of three numbers is $12$. The first plus the second add up to $7$. The second plus the third add up to $9$. Set up a system of three equations with three unknowns and determine the value of these numbers.

In a café, three types of sandwiches are sold: ham, cheese, and omelette. One group of friends orders $3$ ham, $2$ cheese, and $1$ omelette, paying $14$ euros. Another group orders $2$ ham, $1$ cheese, and $3$ omelette, paying $13$ euros. A third group orders $1$ ham, $3$ cheese, and $2$ omelette, paying $15$ euros. Set up and solve a system to calculate the price of each type of sandwich.

Three machines A, B, and C manufacture parts in a factory. Working together, machine A and machine B manufacture $50$ parts per hour. Machine B and machine C together manufacture $70$ parts per hour. Finally, machine A and machine C together manufacture $60$ parts per hour. Calculate how many parts per hour each machine manufactures individually.