Exercises: Systems of three equations with three unknowns

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:

$x - y + z = 2$

$2x + y - z = 1$

$x + 2y + z = 7$

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

$3x + y - z = 0$

$x - 2y + z = 5$

$2x + y + z = 3$

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.

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

$x + y + z = 1$

$2x + y - z = 0$

$3x + 2y = 5$

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

$x - y + 2z = 3$

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

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

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

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.

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.

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

$x + 2y - z = 4$

$2x - y + 3z = 3$

$3x + y + 2z = 7$

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.

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

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

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

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

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

$x + y - z = 2$

$2x - y + z = 1$

$3x + y - z = k$

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.

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.

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 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 $100c + 10d + u$).

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

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

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

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

Geometrically, a linear equation in three unknowns represents a plane in space. Observe the following graphical representation where three planes intersect along a single straight line, forming a sheaf of planes.

Given the system:

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

$\pi_{2}: 2x - y + z = 1$

$\pi_{3}: 4x + 3y - z = k$

Calculate the value of $k$ so that the three planes intersect in a single straight line (a consistent dependent system with one degree of freedom) and express the parametric equation of that line of intersection.

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.

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.

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

$x - y + 2z = 1$

$2x + ay - z = 3$

$3x + y + bz = 4$

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.