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.

$3x + y - z = 0$

$x - 2y + z = 5$

$2x + y + z = 3$

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

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

$x + 2y - z = 4$

$3y + z = 5$

$2z = 4$

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

$x + y = 5$

$y + z = 7$

$x + z = 6$

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 $\lambda$:

$x + 2y - z = 4$

$2x - y + 3z = 3$

$3x + y + 2z = 7$

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

$x + y - z = 2$

$2x - y + z = 1$

$3x + y - z = k$

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

$x - y + 2z = 1$

$2x + ay - z = 3$

$3x + y + bz = 4$

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.

$x + y + z = 1$

$x + my + z = m$

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

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 omelet. One group of friends orders $3$ ham, $2$ cheese, and $1$ omelet, paying $840$ pesos. Another group orders $2$ ham, $1$ cheese, and $3$ omelet, paying $780$ pesos. A third group orders $1$ ham, $3$ cheese, and $2$ omelet, paying $900$ pesos. 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.

The ages of a grandfather, his son, and his grandson sum to $120$ years. The grandfather is twice as old as the son. Furthermore, the son is $30$ years older than the grandson. Determine the age of each person by setting up a system of three equations.

In a basketball game, a player scored a total of $15$ field goals and free throws, getting $30$ points. Knowing that the number of two-point baskets is triple the number of three-point baskets, and that he also scored free throws (which are worth one point), set up a system of equations to find out how many baskets of each type he made.

In a triangle, the sum of its three interior angles $A$, $B$, and $C$ is $180^{\circ}$. Angle $A$ is equal to the sum of angles $B$ and $C$. Furthermore, angle $B$ is twice angle $C$. Set up a system of three equations and determine the measure of the three angles.

The sum of the three digits of a number is $15$. If the order of the digits is reversed, the resulting number is $198$ units greater than the original number. Furthermore, the tens digit is the arithmetic mean of the hundreds digit and the units digit. Find the original number. (Note: A three-digit number $c$, $d$, $u$ has a value of $100c + 10d + u$).

Three alloys of silver, copper, and zinc are available. The first contains $10\%$ silver, $70\%$ copper, and $20\%$ zinc. The second contains $30\%$ silver, $40\%$ copper, and $30\%$ zinc. The third contains $20\%$ silver, $50\%$ copper, and $30\%$ zinc. We want to obtain $100$ kg of a new alloy containing exactly $22\%$ silver and $52\%$ copper. Set up the system of equations to calculate how many kilograms of each alloy must be mixed, and solve it.

A transportation company acquires three types of vehicles: vans, light trucks, and heavy trucks. The total number of vehicles acquired is known to be $30$. The cost of each van is $1\,200\,000$ pesos, each light truck is $1\,800\,000$ pesos, and each heavy truck is $3\,000\,000$ pesos. If the total investment was $51\,000\,000$ pesos, set up the system and show that it is a consistent dependent system. Then, find all possible integer solutions knowing that the company bought at least one vehicle of each type.

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

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

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

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

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.