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:

$x + 2y - z = 4$

$3y + z = 5$

$2z = 4$

Design a linear system of three equations with three unknowns ($x$, $y$, $z$) that simultaneously satisfies these three conditions:

1. It must already be in row echelon (triangular) form, so that it can be solved directly by back-substitution without the need to apply prior transformations.

2. None of the coefficients of the unknowns nor the constant terms can be equal to $0$ or $1$.

3. Its unique solution must be exactly the triple $(2, -1, 3)$.

Explain in a reasoned manner the reverse process you follow to build each equation, from bottom to top (starting with the third and ending with the first), and formally verify that your system leads to that solution.

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:

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

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

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

Solve the following system where one unknown is missing in each equation:

$x + y = 5$

$y + z = 7$

$x + z = 6$

Write a linear system of three equations in three unknowns whose unique solution is the ordered triple $x = 2$, $y = 0$, $z = -1$.

To solve the following system of linear equations:

two students propose different initial strategies:

• Aoife's strategy (direct Gauss): Keep the order of the equations and use the first row ($3x - y + 2z = 7$) directly as the pivot row to create zeros in the column of $x$.

• Niamh's strategy (Gauss with unit pivot): Swap the first and second equations beforehand so that the coefficient of $x$ in the pivot row is $1$, and from there carry out the linear combinations to eliminate the terms in $x$ from the other two rows.

Analyze in a reasoned manner the advantages and disadvantages of each strategy (taking into account the appearance of fractional coefficients and the likelihood of calculation errors). Justify which of the two options is more convenient and apply it step by step until the system is in echelon form and find its solution.

Solve the following non-linear system of equations by applying a suitable change of variables (for example, $u = \frac{1}{x}$, $v = \frac{1}{y}$, $w = \frac{1}{z}$):

$\frac{1}{x} + \frac{1}{y} = 5$

$\frac{1}{y} + \frac{1}{z} = 7$

$\frac{1}{x} + \frac{1}{z} = 6$

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 $\lambda$:

$x + 2y - z = 4$

$2x - y + 3z = 3$

$3x + y + 2z = 7$

Prove algebraically that if a linear system of three equations in three unknowns admits two distinct solutions, for example $S_{1} = (1, 0, 0)$ and $S_{2} = (0, 1, 0)$, then it admits infinitely many solutions. To do this, verify that any linear combination of the form $S = \lambda S_{1} + (1-\lambda)S_{2}$ is also a solution to the system.

Find the values of the parameter $k$ for which the following system is inconsistent:

$x + y - z = 2$

$2x - y + z = 1$

$3x + y - z = k$

Discuss the following system of equations depending on the values of the parameter $m$. State for which values of $m$ the system is consistent independent, consistent dependent, or inconsistent.

$x + y + z = 1$

$x + my + z = m$

$x + y + m^{2}z = m^{2}$

Find the values of the parameters $a$ and $b$ for which the following system of equations is consistent dependent:

$x - y + 2z = 1$

$2x + ay - z = 3$

$3x + y + bz = 4$

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.