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 $x = 2$, $y = 0$, $z = -1$.

Marcos is solving a system of three equations with three unknowns and, after correctly applying the first transformations of the Gaussian elimination method, arrives at the following row echelon system:

In his notebook, Marcos concludes the following:

"Since the last row is $0z = 0$, I solve for $z$ and get $z = \frac{0}{0} = 0$. Substituting into the second row yields $y + 3\cdot 0 = 4 \implies y = 4$. Finally, in the first row we have $x + 2\cdot 4 - 0 = 5 \implies x = -3$. Therefore, the system is consistent and independent (unique solution), and its only solution is the triple $(-3, 4, 0)$".

Identify the conceptual mistake Marcos made when interpreting the row $0z = 0$, explain in detail why his deduction is mathematically incorrect, and correct the exercise: classify the system appropriately and find the general expression of all its solutions in terms of a real parameter $\lambda$.

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.

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.

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.

To solve the following system of linear equations:

two students propose different initial strategies:

• Marta'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 $x$.

• Irene'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.

An urgent courier company in Seville renews its delivery fleet by acquiring a total of $20$ vehicles across three low-emission technologies: pure electric vans ($x$), plug-in hybrid vans ($y$), and compressed natural gas vans ($z$). The following information about the purchase is available:

• The average acquisition cost is €$45\,000$ per electric van, €$35\,000$ per hybrid, and €$25\,000$ per natural gas van, reaching a total investment of €$670\,000$.

• Due to internal environmental policy, the number of electric vans acquired must be exactly equal to the sum of the number of hybrids plus twice the number of natural gas vans.

Set up the system of three linear equations with three unknowns that models the situation, solve it using Gaussian elimination, and interpret the result indicating whether the solution obtained is feasible and how many vehicles of each type make up the new fleet.

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 transport 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 $20.000$ euros, each light truck is $30.000$ euros, and each heavy truck is $50.000$ euros. If the total investment was $850.000$ euros, 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.

A wine cooperative in La Rioja prepares a special batch of $1000$ liters by blending three grape varieties: Tempranillo ($x$), Garnacha ($y$), and Graciano ($z$). Winemaking costs are €$4$ per liter of Tempranillo, €$2$ per liter of Garnacha, and €$7$ per liter of Graciano. The total budget allocated for the production of the entire batch is €$3900$. In addition, the winemaker sets as a tasting criterion that the volume of Tempranillo must be exactly twice the sum of the volumes of Garnacha and Graciano combined.

1. Estimate and bound: Without solving the system formally, deduce the exact volume of Tempranillo that the batch must contain from the winemaker's constraint and the total volume. Next, reason whether it is possible for the most expensive variety (Graciano) to exceed $200$ liters, comparing the remaining cost with the prices of Garnacha and Graciano.

2. Solve and verify: Set up the $3\times 3$ linear system, transform it into a triangular system using Gaussian elimination, and find the required liters of each variety. Check whether the values obtained match your initial deductions and whether the resulting solution makes physical sense for the winemaker.

Consider the following system of linear equations with unknowns $x$, $y$, $z$, involving two real parameters $k$ and $m$:

Each of these three equations represents a plane in three-dimensional space.

1. Apply Gaussian elimination until the system is transformed into an echelon form where the last equation has the form $a(k)\cdot z = b(m)$, clearly detailing each elementary row operation.

2. Determine the numerical values that the parameters $k$ and $m$ must take so that the three planes intersect in a common line (consistent dependent system). For those values, find the general solution in terms of a parameter $\lambda$.

3. Reasonably determine what condition $k$ and $m$ must satisfy for the system to be inconsistent. In that case, can the three planes intersect at a single point? Justify your answer by connecting the algebraic result obtained in the last row with the geometric interpretation of the planes.