Gaussian 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:

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

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

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

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:

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

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 $27$ pesos. How much money does each person have?"

In a sporting goods store, three types of balls are sold: soccer balls, basketballs, and volleyballs. One school buys $2$ soccer balls, $1$ basketballs, and $1$ volleyballs for $1\,600$ pesos. Another school buys $1$ soccer balls, $2$ basketballs, and $1$ volleyballs for $1\,700$ pesos. A third school buys $1$ soccer balls, $1$ basketballs, and $2$ volleyballs for $1\,500$ pesos. Set up a system of equations and solve it using Gaussian elimination to calculate the price of each type of ball.

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 $C(2, 7)$.

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 $100$ liters 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 liters of each solution must be mixed.

A company invests a total of $400\,000$ pesos in three different investment funds, which offer annual returns of $4\%$, $5\%$, and $6\%$. After one year, the total interest generated amounts to $19\,600$ pesos. 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.

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.

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

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.

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.

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

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

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

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.