Solving systems by elimination

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

$\left\{ \begin{array}{rcl} x + 2y - z & = & 4 \\ 3y + z & = & 5 \\ 2z & = & 4 \end{array} \right.$

Solve the following system in which some unknowns are missing from the equations, reordering them appropriately if you consider it necessary:

$\left\{ \begin{array}{rcl} x + y & = & 3 \\ y + z & = & 5 \\ x + z & = & 4 \end{array} \right.$

Find the value of the coefficient $a$ knowing that the triple $(1, -1, 2)$ is a solution to the following system:

$\left\{ \begin{array}{rcl} x + y + z & = & 2 \\ 2x - y - z & = & 1 \\ x + ay + 2z & = & 8 \end{array} \right.$

Apply Gaussian elimination to triangularise and solve the following system of equations:

$\left\{ \begin{array}{rcl} x + y + z & = & 6 \\ x - y + 2z & = & 5 \\ x + y - z & = & 0 \end{array} \right.$

Solve the following system using Gaussian elimination, eliminating the variable $x$ from the second and third equations first:

$\left\{ \begin{array}{rcl} x + 2y + z & = & 7 \\ 2x - y + 3z & = & 11 \\ -x + y - z & = & -4 \end{array} \right.$

Aoife tries to put the following system of linear equations into row echelon form using the Gaussian elimination method:

In the first step, she eliminates the variable $x$ from the second and third equations by doing $E_2 \leftarrow E_2 - 2E_1$ and $E_3 \leftarrow E_3 - 3E_1$, correctly obtaining:

For the second step, in order to eliminate the variable $y$ from the third row, Aoife decides to replace $E_3$ with the combination $E_3 \leftarrow 3E_3 - 5E_1$.

1. a) Explain in detail why the operation proposed by Aoife is a conceptual error in the Gaussian elimination method, and indicate what unwanted effect it produces on the row echelon form of the system.
2. b) Write the correct elementary row operation that Aoife should have applied to $E_3$ using the second equation.
3. c) Apply this correction, complete the row echelon form, and classify the system according to its number of solutions.

Imagine you are faced with the following system of three equations with three unknowns:

To apply the Gaussian elimination method, the first objective is to set a pivot in the first position of the first row. You are presented with three starting alternatives:

• Strategy 1: Keep the order of the equations and operate with cross-multiples ($E_2 \leftarrow 3E_2 - 2E_1$ and $E_3 \leftarrow 3E_3 - E_1$).

• Strategy 2: Swap the first and third rows ($E_1 \leftrightarrow E_3$) so that the coefficient of $x$ is $1$.

• Strategy 3: Multiply the first row by $\frac{1}{3}$ ($E_1 \leftarrow \frac{1}{3}E_1$) to simplify it before pivoting.

1. a) Compare the three strategies, reasoning which one is the most advantageous for minimising arithmetic operations and avoiding large numbers or awkward fractions.
2. b) Choose the strategy you considered optimal and solve the system by putting it into row echelon form using Gauss and applying back-substitution.

Solve the following system of linear equations, operating first with the fractions to obtain an equivalent system with integer coefficients:

$\left\{ \begin{array}{rcl} \frac{x}{2} + \frac{y}{3} - z & = & 1 \\ x - \frac{y}{2} + \frac{z}{3} & = & 2 \\ \frac{x}{3} + y - \frac{z}{2} & = & 3 \end{array} \right.$

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

$\left\{ \begin{array}{rcl} \frac{1}{x} + \frac{2}{y} - \frac{1}{z} & = & 2 \\ \frac{2}{x} - \frac{1}{y} + \frac{3}{z} & = & 9 \\ \frac{3}{x} + \frac{3}{y} - \frac{2}{z} & = & 3 \end{array} \right.$

Without solving it completely, classify the following echelon system (as a consistent independent system, consistent dependent system, or inconsistent system) and justify your answer:

$\left\{ \begin{array}{rcl} x - 3y + 2z & = & 1 \\ 2y - z & = & 4 \\ 0z & = & 7 \end{array} \right.$

Write the solution to the following echelon system in terms of a parameter $\lambda$:

$\left\{ \begin{array}{rcl} x + y + z & = & 3 \\ y - 2z & = & 2 \\ 0z & = & 0 \end{array} \right.$

Solve the following system containing four equations and three unknowns. State whether any equation is redundant or if the system is inconsistent:

$\left\{ \begin{array}{rcl} x - y + z & = & 2 \\ 2x + y - z & = & 1 \\ x + 2y - 2z & = & -1 \\ 3x + y + z & = & 5 \end{array} \right.$

Discuss the consistency of the following system of linear equations depending on the parameter $k$ and solve it for the case in which it is a consistent dependent system:

$\left\{ \begin{array}{rcl} x + y - z & = & 1 \\ 2x - y + z & = & 2 \\ x - 2y + 2z & = & k \end{array} \right.$

Study and discuss the following system of linear equations depending on the values of the parameter $m$. Solve it for $m = 2$.

$\left\{ \begin{array}{rcl} x + y + z & = & 1 \\ 2x + my + z & = & 2 \\ x + y + mz & = & 3 \end{array} \right.$

Discuss the following system and find the value or values of $a$ for which the system is consistent dependent. Write the general solution for that case.

$\left\{ \begin{array}{rcl} x + y - z & = & 10 \\ 2x - y + 3z & = & 20 \\ 5x - y + 5z & = & a \end{array} \right.$

You want to design an exam task about the Gaussian elimination method with discussion of solutions. To do this, you start from two fixed equations:

You want to add a third equation of the form $3x + k\times y - z = c$ (where $k$ and $c$ are real numbers) such that, when triangulating the system using standard linear combinations, the coefficients of all three unknowns in the third equation vanish simultaneously ($0x + 0y + 0z$).

1. a) Apply the transformations of the Gaussian elimination method treating $k$ and $c$ as constants, and determine the unique numerical value that $k$ must take so that the coefficients of $y$ and $z$ vanish at the same time in the third row.
2. b) Find the expression left in the constant term after this step, and deduce what value $c$ must have for the system to be consistent dependent.
3. c) Write the general solution of the consistent dependent system in terms of a parameter $\lambda \in \mathbb{R}$, and justify what would happen geometrically to the three planes represented by the equations if $c$ took any value other than the one obtained.

Consider the following system of linear equations depending on two real parameters $a$ and $b$, where each equation represents a plane in three-dimensional space:

1. a) Triangulate the system using the Gaussian elimination method keeping the parameters $a$ and $b$. From the resulting echelon equation in the last row, classify the system with reasoning (consistent independent, consistent dependent, or inconsistent) according to the different values that $a$ and $b$ can take.
2. b) For the case in which the system is consistent dependent, solve it by expressing the general solution in terms of a parameter $\lambda \in \mathbb{R}$, and geometrically interpret the relative position of the three planes.
3. c) A student states: “If we set $a = 2$ and $b = 7$, the system is inconsistent, which necessarily proves that the third plane is parallel to one of the first two.” Analyse the truth or falsity of this statement by examining the coefficients of the original equations, and describe the actual geometric configuration of the three planes in that case.

Geometrically, a linear equation with three unknowns represents a plane in space. Solve the system formed by the planes $\pi_1$, $\pi_2$ and $\pi_3$, interpret the result you obtain and state what geometric figure they form at their intersection.

$\left\{ \begin{array}{rcl} x + y + z & = & 3 \\ 2x - y + z & = & 2 \\ 3x + 2z & = & 5 \end{array} \right.$

Prove, using Gaussian elimination, that the system of equations:

$\left\{ \begin{array}{rcl} x + y + z & = & a \\ x - y + z & = & b \\ 2x + 2z & = & c \end{array} \right.$

is consistent only if the relation $a + b = c$ holds. Explain the algebraic meaning of this condition.

Set up a system of three equations in three unknowns and solve it using Gaussian elimination: "Kenia has three times as much money as Berta. Berta has twice as much money as Ionut. Together, the three of them have $27$ euros. How much money does each person have?"

In a sports equipment shop, three types of balls are sold: footballs, basketballs, and volleyballs. One school buys $2$ footballs, $1$ basketballs, and $1$ volleyballs for $80$ euros. Another school buys $1$ footballs, $2$ basketballs, and $1$ volleyballs for $85$ euros. A third school buys $1$ footballs, $1$ basketballs, and $2$ volleyballs for $75$ euros. Set up a system of equations and solve it using Gaussian elimination to calculate the price of each type of ball.

The sum of the ages of a grandfather, his son, and his grandson is $190$ years. The grandfather is three times as old as the grandson. Furthermore, in $10$ years' time, the father's age will be twice the age the grandson will have at that time. Set up a system of equations and find the current ages of all three.

In a chemistry laboratory, three solutions of the same acid with concentrations of $10\%$, $20\%$, and $50\%$ are available. We wish to obtain a mixture of $100$ litres with a concentration of $30\%$. Knowing that the volume used of the solution at $10\%$ must equal the volume of the solution at $50\%$, set up and solve a system using Gaussian elimination to determine how many litres of each solution must be mixed.

A company invests a total of $20000$ euros in three different investment funds, which offer annual returns of $4\%$, $5\%$, and $6\%$. After one year, the total interest generated amounts to $980$ euros. If the money invested in the fund at $5\%$ is double the money invested in the fund at $6\%$, calculate how much money was allocated to each fund using Gaussian elimination.

In a basketball game, a team scored a total of $80$ points through $42$ baskets (some worth $1$ point, some $2$ points, and some $3$ points). Knowing that the number of $2$-point baskets is equal to the sum of the $1$- and $3$-point baskets plus $6$, find the number of baskets of each type.

Find the coefficients $a$, $b$, and $c$ of the polynomial $P(x) = ax^2 + bx + c$ knowing that its graph passes through the points $A(-1, 4)$, $B(1, 2)$, and $C(2, 7)$.

The sum of the three digits of a number is $15$. If the order of the digits is reversed, the number obtained is $198$ units greater than the original number. Furthermore, the tens digit is known to be the arithmetic mean of the hundreds digit and the units digit. Set up a system of equations and find the original number.

Three machines, $A$, $B$, and $C$, work on production in a factory. If $A$ and $B$ work together, they complete a batch in $12$ hours. If $B$ and $C$ work together, they complete it in $20$ hours. If $A$ and $C$ work together, they complete it in $15$ hours. Set up a system of linear equations (hint: use as unknowns the fraction of work each machine completes in one hour) and calculate how long each machine would take to complete the batch working alone.

On an Irish Rail intercity train between Dublin and Cork, $60$ people are travelling in a fully occupied passenger car, divided across three ticket types: Basic ($x$), Standard ($y$), and Premium ($z$).

The total revenue from the car amounts to €$2,550$. The prices of each ticket type are: €$30$ for the Basic ticket, €$45$ for the Standard ticket, and €$70$ for the Premium ticket.

Furthermore, the company reports that the number of passengers travelling with a Basic ticket is exactly double the sum of those travelling with the other two tickets combined.

1. a) Model the situation by setting up a system of three linear equations with three unknowns.
2. b) Triangulate the system using the Gaussian elimination method, clearly detailing each of the row operations performed.
3. c) Solve the row echelon system and check whether the values obtained are non-negative integers consistent with the real-world context of the problem.

A pharmaceutical laboratory in Ireland must prepare exactly $100\text{ ml}$ of a compound by mixing three liquid substances: $A$, $B$, and $C$. Three conditions must be satisfied:

1. The combined volume of the three substances must be $100\text{ ml}$.

2. The cost per millilitre of each substance is €$1.50$ for $A$, €$2.00$ for $B$, and €$4.50$ for $C$. The exact total budget allocated to the mixture is €$230$ (which represents an overall average cost of €$2.30$/ml).

3. For chemical stability, the volume of substance $C$ must be exactly one fourth of the sum of the volumes of $A$ and $B$.

1. a) Prior estimation phase: Algebraically deduce the exact volume that substance $C$ must have by directly relating conditions 1 and 3 without yet needing to set up the Gaussian elimination method. Knowing that the average cost of the mixture is €$2.30$/ml and that $C$ is the most expensive component (€$4.50$/ml), reasonably estimate whether the rest of the mixture must contain a greater volume of substance $A$ (€$1.50$/ml) or of substance $B$ (€$2.00$/ml).
2. b) Calculation phase: Model the problem using a $3 \times 3$ linear system, formally triangulate it using the Gaussian elimination method specifying each row operation, and obtain the exact volumes of each liquid.
3. c) Verification phase: Compare the result of the system with your estimates from part a) and assess the practical consistency of the solution found.