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 needing to solve it completely, classify the following echelon system (as consistent and determinate, consistent and indeterminate, or inconsistent) 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 triplet $(1, -1, 2)$ 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, first eliminating the variable $x$ from the second and third equations:

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

Solve the following system in which some unknowns are missing from the equations, ordering 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 with three unknowns and solve it using Gaussian elimination: "Ashley has three times as much money as Bethany. Bethany has twice as much money as Daniel. Between the three of them, they have $\$27$. How much money does each one have?"

A sports equipment shop sells three types of balls: soccer balls, basketballs, and volleyballs. A school buys $2$ soccer balls, $1$ basketballs, and $1$ volleyballs for $\$80$. Another school buys $1$ soccer balls, $2$ basketballs, and $1$ volleyballs for $\$85$. A third school buys $1$ soccer balls, $1$ basketballs, and $2$ volleyballs for $\$75$. 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, first operating 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 are available with concentrations of $10\%$, $20\%$, and $50\%$. It is desired to obtain a mixture of $100$ liters with a concentration of $30\%$. Knowing that the volume used of the $10\%$ solution must be equal to the volume of the $50\%$ solution, 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 the age of the grandson. Furthermore, it is known that in $10$ years, the father's age will be twice the age the grandson will be 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. Indicate if there is any redundant equation 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 $\$20000$ in three different investment funds, which offer annual returns of $4\%$, $5\%$, and $6\%$. After one year, the total interest generated amounts to $\$980$. If the money invested in the $5\%$ fund is twice that invested in the $6\%$ fund, calculate what amount was allocated to each fund using Gaussian elimination.

In a basketball game, a team has scored a total of $80$ points through $42$ baskets (some worth $1$ point, others $2$ points, and others $3$ points). Knowing that the number of $2$-point baskets is equal to the sum of the $1$- and $3$-point baskets plus $5$, find the number of baskets of each type.

Discuss the consistency of the following system of linear equations in terms of the parameter $k$ and solve it for the case where it is consistent and indeterminate:

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

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

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

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 obtained, and indicate 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 in terms of 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, it is known that the tens digit is 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 in the production of a factory. If $A$ and $B$ work together, they finish a batch in $12$ hours. If $B$ and $C$ work, they finish it in $20$ hours. If $A$ and $C$ work, they finish it in $15$ hours. Set up a system of linear equations (hint: use as unknowns the fraction of work that each machine does in one hour) and calculate how long each machine would take to do the batch working alone.

Discuss the following system and find the value or values of $a$ for which the system is consistent and indeterminate. 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 only consistent if the relationship $a + b = c$ is met. Explain the algebraic meaning of this condition.