Gaussian Elimination
Emma, Malik, and Sofia each buy a snack at the school stand, but a banner covers the price board. Emma's order (1 pretzel, 1 drink, 1 popcorn) cost $\$9$. Malik's (2 pretzels, 1 drink, 1 popcorn) cost $\$12$. Sofia's (1 pretzel, 2 drinks, 1 popcorn) cost $\$11$. Can you find the price of each item? That is three equations with three unknowns, which is too much to juggle at once. Gaussian elimination turns the system, step by step, into an equivalent system (same solution) that is easy to read from the bottom up.
Echelon form and back substitution
The bottom equation has only one unknown, so you can solve it right away. Then you work upward, plugging each answer into the equation above. This is called back substitution.
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:
$\left\{ \begin{array}{rcl} x + 2y - z & = & 4 \\ 3y + z & = & 5 \\ 2z & = & 4 \end{array} \right.$
Row operations that keep the solution
How do we get to echelon form? By changing the equations in ways that never change the solution.
We write $E_2 \to E_2 - 2E_1$ to mean "replace equation 2 with equation 2 minus 2 times equation 1." Some problems call these elementary operations. The goal is always to create zeros below the staircase.
Checking a solution
A solution $(x, y, z)$ must make every equation true. That gives a quick way to check an answer, or to find a missing coefficient.
Find the value of the coefficient $a$ knowing that the triplet $(1, -1, 2)$ is a solution to the following system:…
Find the value of the coefficient $a$ knowing that the triplet $(1, -1, 2)$ is a solution to the following system:
$\left\{ \begin{array}{rcl} x + y + z & = & 2 \\ 2x - y - z & = & 1 \\ x + ay + 2z & = & 8 \end{array} \right.$
Triangulating a 3 × 3 system
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, first eliminating the variable $x$ from the second and third equations:…
Solve the following system using Gaussian elimination, first eliminating the variable $x$ from the second and third equations:
$\left\{ \begin{array}{rcl} x + 2y + z & = & 7 \\ 2x - y + 3z & = & 11 \\ -x + y - z & = & -4 \end{array} \right.$
Fractions and missing unknowns
Not every system arrives in neat form. Two quick fixes handle most cases.
Fractions. Multiply each equation (both sides!) by the least common multiple of its denominators. That gives an equivalent system with integer coefficients. For example, $\frac{x}{2}+\frac{y}{4}=2$ times $4$ becomes $2x+y=8$.
Missing unknowns. If a variable does not appear, its coefficient is $0$. Writing the zero explicitly keeps the columns aligned. If $E_1$ is missing $x$, swap it with an equation that has $x$.
Solve the following system in which some unknowns are missing from the equations, ordering them appropriately if you consider it necessary:…
Solve the following system in which some unknowns are missing from the equations, ordering them appropriately if you consider it necessary:
$\left\{ \begin{array}{rcl} x + y & = & 3 \\ y + z & = & 5 \\ x + z & = & 4 \end{array} \right.$
Solve the following system of linear equations, first operating with the fractions to obtain an equivalent system with integer coefficients:…
Solve the following system of linear equations, first operating with the fractions to obtain an equivalent system with integer coefficients:
$\left\{ \begin{array}{rcl} \frac{x}{2} + \frac{y}{3} - z & = & 1 \\ x - \frac{y}{2} + \frac{z}{3} & = & 2 \\ \frac{x}{3} + y - \frac{z}{2} & = & 3 \end{array} \right.$
Classifying a system: one, infinitely many, or none
While eliminating, the last equation can do something surprising: all the unknowns can disappear.
Some problems use the words determinate and indeterminate for independent and dependent. It is the same classification.
Name the free unknown with the letter the problem uses. It is often $t$, and sometimes $\lambda$.
More equations than unknowns
A system can have four equations and three unknowns. Eliminate exactly as before. Each extra equation will either turn into $0=0$ (it was redundant, a combination of the others) or into $0=k\neq0$ (the system is inconsistent). Only after all the zero rows are accounted for do you read off the solution.
Without needing to solve it completely, classify the following echelon system (as consistent and determinate, consistent and indeterminate, or inconsistent) and justify…
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 $\lambda$:…
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. Indicate if there is any redundant equation or if the system is inconsistent:…
Solve the following system containing four equations and three unknowns. Indicate if there is any redundant equation or if the system is inconsistent:
$\left\{ \begin{array}{rcl} x - y + z & = & 2 \\ 2x + y - z & = & 1 \\ x + 2y - 2z & = & -1 \\ 3x + y + z & = & 5 \end{array} \right.$
Systems with a parameter
Sometimes a system contains a letter besides the unknowns, such as $k$ or $m$. The answer can depend on its value.
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:…
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:
$\left\{ \begin{array}{rcl} x + y - z & = & 1 \\ 2x - y + z & = & 2 \\ x - 2y + 2z & = & k \end{array} \right.$
Study and discuss the following system of linear equations in terms of the values of the parameter $m$. Solve it for $m = 2$.…
Study and discuss the following system of linear equations in terms of the values of the parameter $m$. Solve it for $m = 2$.
$\left\{ \begin{array}{rcl} x + y + z & = & 1 \\ 2x + my + z & = & 2 \\ x + y + mz & = & 3 \end{array} \right.$
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.…
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.
$\left\{ \begin{array}{rcl} x + y - z & = & 10 \\ 2x - y + 3z & = & 20 \\ 5x - y + 5z & = & a \end{array} \right.$
Three planes and a consistency condition
| What the planes do | Elimination gives | Solutions |
|---|---|---|
| Meet at one point | a lone unknown at the bottom | one |
| Share a common line | a row $0=0$ | infinitely many |
| Three coincident planes | two rows of $0=0$ | infinitely many (a plane) |
| No point in common | a row $0=k\neq0$ | none |
A condition on the constants
If the left side of one equation is a combination of the other left sides, that equation's right side must be the same combination of the other right sides. If it is not, the system is inconsistent.
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 obtained, and indicate what geometric figure they form at their intersection.
$\left\{ \begin{array}{rcl} x + y + z & = & 3 \\ 2x - y + z & = & 2 \\ 3x + 2z & = & 5 \end{array} \right.$
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 only consistent if the relationship $a + b = c$ is met. Explain the algebraic meaning of this condition.
Changing variables
Some systems are not linear in $x$, $y$, $z$ but become linear after a change of variables. A common case is when the unknowns appear as reciprocals.
Find the solutions to the following non-linear system of equations by first making a suitable change of variables ($u = \frac{1}{x}$, $v = \frac{1}{y}$,…
Find the solutions to the following non-linear system of equations by first making a suitable change of variables ($u = \frac{1}{x}$, $v = \frac{1}{y}$, $w = \frac{1}{z}$):
$\left\{ \begin{array}{rcl} \frac{1}{x} + \frac{2}{y} - \frac{1}{z} & = & 2 \\ \frac{2}{x} - \frac{1}{y} + \frac{3}{z} & = & 9 \\ \frac{3}{x} + \frac{3}{y} - \frac{2}{z} & = & 3 \end{array} \right.$
Setting up word problems
Elimination does the hard algebra, but the setup is the real skill. Follow the same five steps every time.
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…
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…
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.
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…
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.
Mixtures, investments, and counting problems
Many three-unknown problems share one pattern: amount = rate × quantity. You write one equation for the total quantity and one for the total amount.
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 $100$…
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.
A company invests a total of $\$20000$ in three different investment funds, which offer annual returns of $4\%$, $5\%$, and $6\%…
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…
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.
Curves, digits, and work rates
Three more situations that produce $3\times3$ systems.
Fitting a parabola. A point $(x_0, y_0)$ on $y=ax^2+bx+c$ gives the equation $a x_0^2+bx_0+c=y_0$. Each point gives one equation in $a$, $b$, $c$, and three points give three equations.
Slide the sliders to see how $a$, $b$, and $c$ move the parabola.
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 $C(2, 7)$.
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, 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…
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.