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 triangularise and solve the following system of equations:

$x - y + z = 2$

$2x + y - z = 1$

$x + 2y + z = 7$

Check if the triplet $x = 1$, $y = -1$, $z = 2$ 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 sum to $7$. The second plus the third sum to $9$. Set up a system of three equations with three unknowns and determine the value of these numbers.

Triangularise the following system using Gaussian elimination and indicate if 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 verify that it is a consistent and indeterminate 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 with three unknowns whose only solution is the triplet $x = 2$, $y = 0$, $z = -1$.

A cafe sells three types of sandwiches: ham, cheese, and Spanish omelet. A group of friends orders $3$ ham, $2$ cheese, and $1$ Spanish omelet sandwiches, paying $\$14$. Another group orders $2$ ham, $1$ cheese, and $3$ Spanish omelet sandwiches, paying $\$13$. A third group orders $1$ ham, $3$ cheese, and $2$ Spanish omelet sandwiches, paying $\$15$. 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 exactly twice as old as the son. Furthermore, the son is $30$ years older than the grandson. Determine the age of each of them by setting up a system of three equations.

Solve the following consistent and indeterminate system and express its infinite 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 has scored a total of $15$ baskets, getting $31$ points. Knowing that the number of two-point baskets is three times the number of three-point baskets, and that they have also scored free throws (which are worth one point), set up a system of equations to find out how many baskets of each type they have scored.

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, machines A and B manufacture $50$ parts per hour. Machines B and C together manufacture $70$ parts per hour. Finally, machines A and 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 in terms of the values of the parameter $m$. Indicate for which values of $m$ the system is consistent and determinate, consistent and indeterminate, 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 and units digits. 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 making a suitable change of variable (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 with three unknowns represents a plane in space. Observe the following graphical representation where three planes intersect along a single line, forming a pencil 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 line (consistent and indeterminate system with one degree of freedom) and express the parametric equation of this line of intersection.

A transport company acquires three types of vehicles: vans, light trucks, and heavy trucks. It is known that the total number of vehicles acquired is $30$. The cost of each van is $\$20.000$, that of each light truck is $\$30.000$, and that of each heavy truck is $\$50.000$. If the total investment has been $\$850.000$, set up the system and show that it is a consistent and indeterminate system. Then, find all possible integer solutions knowing that the company has 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 $100$ pounds of a new alloy containing exactly $22\%$ silver and $52\%$ copper. Set up the system of equations to calculate how many pounds of each alloy we must mix and solve it.

Find the values of the parameters $a$ and $b$ so that the following system of equations is consistent and indeterminate:

$x - y + 2z = 1$

$2x + ay - z = 3$

$3x + y + bz = 4$

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