Gauss's Method
At a corner store (colmado), three customers buy sodas, juices, and little bottles of water in different amounts, and each pays a different total. How much does each product cost? There are three unknowns and three equations, and you can't solve it by eye. Gauss's method transforms the system, step by step, into another equivalent one (with the same solution) but much easier: one where each equation has one fewer unknown than the one before. In the end you only have to read off the result from the bottom up.
An echelon system is solved from the bottom up
The last equation has just one unknown, so you solve for it directly. You substitute that value into the equation above it, and so on. This is back substitution.
If a system is missing unknowns in some equations, always write the unknowns in the same order $x, y, z$ and leave gaps with coefficient $0$.
Solve the following system of linear equations, which is already in echelon form, using back-substitution:…
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:…
Solve the following system in which some unknowns are missing from the equations, reordering them appropriately if you consider it necessary:
$\left\{ \begin{array}{rcl} x + y & = & 3 \\ y + z & = & 5 \\ x + z & = & 4 \end{array} \right.$
Transformations that don't change the solution
These are the same elimination operations you already know from $2\times 2$ systems, applied in a fixed order: they are used to leave zeros below the diagonal, equation by equation.
A quick check: if an ordered triple is a solution of a system, then when you substitute it, it must satisfy all the equations. This is useful for finding an unknown coefficient.
Find the value of the coefficient $a$ knowing that the triple $(1, -1, 2)$ is a solution to the following system:…
Find the value of the coefficient $a$ knowing that the triple $(1, -1, 2)$ is a solution to the following system:
$\left\{ \begin{array}{rcl} x + y + z & = & 2 \\ 2x - y - z & = & 1 \\ x + ay + 2z & = & 8 \end{array} \right.$
Triangularize a 3×3 system step by step
The recipe has two phases. First, use $E_1$ to eliminate $x$ from $E_2$ and $E_3$. Then use $E_2$ to eliminate $y$ from $E_3$. With that the system is in echelon form.
Apply Gaussian elimination to triangularise and solve the following system of equations:…
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:…
Solve the following system using Gaussian elimination, eliminating the variable $x$ from the second and third equations first:
$\left\{ \begin{array}{rcl} x + 2y + z & = & 7 \\ 2x - y + 3z & = & 11 \\ -x + y - z & = & -4 \end{array} \right.$
Classify the system without finishing the solution
When you put the system in echelon form, something unexpected can happen in the last equation: all the unknowns can disappear.
Without solving it completely, classify the following echelon system (as a consistent independent system, consistent dependent system, or inconsistent system) and…
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 $\lambda$:…
Write the solution to the following echelon system in terms of a parameter $\lambda$:
$\left\{ \begin{array}{rcl} x + y + z & = & 3 \\ y - 2z & = & 2 \\ 0z & = & 0 \end{array} \right.$
Discuss a system with a parameter
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…
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 $m = 2$.…
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.…
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.
$\left\{ \begin{array}{rcl} x + y - z & = & 10 \\ 2x - y + 3z & = & 20 \\ 5x - y + 5z & = & a \end{array} \right.$
Set up a problem with three unknowns
Most problems solved with Gauss's method boil down to translating the statement into equations. The steps are always the same.
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…
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 $1$…
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.
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 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.
Other models: points, mixtures, investments, and baskets
Many different problems are translated the same way: each piece of data in the statement is a linear equation.
Polynomial through three points. If the graph of $P(x)=ax^2+bx+c$ passes through a point $(x_0, y_0)$, then $P(x_0)=y_0$. Each point gives an equation in $a$, $b$, and $c$.
Mixtures and investments. These use percentages: $5\%$ of a quantity $y$ is $0.05\,y$. One equation is usually the total (liters, pesos) and another is the result (pure acid, interest). A condition like "twice as much as" or "equal to" gives the third equation.
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 $C(2, 7)$.
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…
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 $1\,200\,000$ pesos in three different investment funds, which offer annual returns of $4\%$, $5\%$, and $6\%$. After one year, the total…
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.
Digit problems and work problems
Two types of problems need a setup trick.
Digits. A three-digit number $\overline{abc}$ is worth $100a+10b+c$. When the digits are reversed it is worth $100c+10b+a$. The difference is $99(c-a)$, which simplifies things a lot.
Work. If a machine takes $t$ hours alone, in one hour it does the fraction $\frac{1}{t}$ of the batch. If two machines together take $12$ hours, in one hour they do $\frac{1}{12}$ of the batch between the two. You take the rate (fraction per hour) of each one as the unknown, and at the end you invert it.
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…
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.
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.…
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…
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.
Special cases: fractions, change of variable, and extra equations
Fractional coefficients. Multiply each equation by the least common multiple of its denominators. For example, $\frac{x}{2}+\frac{y}{3}=4$ multiplied by $6$ is $3x+2y=24$. You get an equivalent system with integer coefficients.
Change of variable. If the unknowns appear as $\frac{1}{x}$, $\frac{1}{y}$, $\frac{1}{z}$, let $u=\frac{1}{x}$, $v=\frac{1}{y}$, $w=\frac{1}{z}$. The system becomes linear in $u$, $v$, $w$. You solve it and then undo the change.
More equations than unknowns. With four equations and three unknowns you triangularize the same way. When eliminating, some equation may become $0=0$ (it was redundant: it followed from the others) or $0=k\ne0$ (the system is inconsistent).
Compatibility condition. If in a system the constant terms are letters ($a$, $b$, $c$), when you triangularize the last equation may become $0=$ an expression with those letters. Then the system has a solution only when that expression equals $0$: this is the compatibility condition. It means that one equation is a combination of the others and its constant term must be the same combination; otherwise it contradicts them.
Solve the following system of linear equations, operating first with the fractions to obtain an equivalent system with integer coefficients:…
Solve the following system of linear equations, operating first with the fractions to obtain an equivalent system with integer coefficients:
$\left\{ \begin{array}{rcl} \frac{x}{2} + \frac{y}{3} - z & = & 1 \\ x - \frac{y}{2} + \frac{z}{3} & = & 2 \\ \frac{x}{3} + y - \frac{z}{2} & = & 3 \end{array} \right.$
Solve the following system containing four equations and three unknowns. State whether any equation is redundant or if the system is inconsistent:…
Solve the following system containing four equations and three unknowns. State whether any equation is redundant or if the system is inconsistent:
$\left\{ \begin{array}{rcl} x - y + z & = & 2 \\ 2x + y - z & = & 1 \\ x + 2y - 2z & = & -1 \\ 3x + y + z & = & 5 \end{array} \right.$
Find the solutions to the following non-linear system of equations by applying a suitable change of variables beforehand ($u = \frac{1}{x}$, $v = \frac{1}{y}$,…
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…
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:…
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.