Exercises: Gaussian elimination

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

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

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

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

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 $\lambda$:

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

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

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

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

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

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 $4\,800$ pesos. Another school buys $1$ soccer balls, $2$ basketballs, and $1$ volleyballs for $5\,100$ pesos. A third school buys $1$ soccer balls, $1$ basketballs, and $2$ volleyballs for $4\,500$ pesos. Set up a system of equations and solve it using Gaussian elimination to calculate the price of each type of ball.

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 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.

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.

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

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

A company invests a total of $1\,200\,000$ pesos in three different investment funds, which offer annual returns of $4\%$, $5\%$, and $6\%$. After one year, the total interest generated amounts to $58\,800$ 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.

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.

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:

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

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

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

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.

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

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

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

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.

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.