What a system is really asking

A linear equation is an equation where each unknown appears only to the first power, and no unknown is multiplied by another unknown. For example,

$$ax+by=c$$

Here $x$ and $y$ are unknown quantities, $a$ and $b$ are coefficients multiplying those unknowns, and $c$ is the constant term. The units of $x$, $y$ and $c$ depend on the context; $a$ and $b$ take whatever units are needed to make the equation dimensionally consistent.

A system of linear equations is a set of linear equations that all have to be true at the same time. A solution is an ordered pair or ordered triple of values that satisfies every equation in the system simultaneously. In this topic, the IB limit is a maximum of three equations in three unknowns, so you are working with systems such as

$$\begin{cases} ax+by=c\\ dx+ey=f \end{cases}$$

where $d$ and $e$ are further coefficients and $f$ is a further constant term, or

$$\begin{cases} ax+by+cz=d\\ ex+fy+gz=h\\ ix+jy+kz=l \end{cases}$$

where $z$ is a third unknown, $g$, $i$, $j$ and $k$ are further coefficients, and $h$ and $l$ are further constant terms.

Geometrically, $ax+by=c$ represents a straight line in the $xy$-plane. A two-variable system is asking where the lines meet. In the same way, $ax+by+cz=d$ represents a plane in three-dimensional space, so a three-variable system asks where the planes meet. The intersection might be one point, no point, or infinitely many points.

Three possible outcomes

A system is consistent if it has at least one solution. A system is inconsistent if it has no solution. A system has a unique solution if exactly one ordered pair or ordered triple satisfies all equations. It has infinitely many solutions if a whole line, plane, or coincident object satisfies the system.

For two equations in two unknowns, the picture helps a lot:

• two non-parallel lines meet once, giving a unique solution;
• parallel distinct lines do not meet, giving no solution;
• the same line written twice gives infinitely many solutions.

For three equations in three unknowns, use planes rather than lines. Three planes can meet in one point, fail to share a common point, or share a whole line or plane. The algebra tells you which case you have.

Algebraic solution by elimination

Elimination is an algebraic method where equations are replaced by equivalent equations so that one unknown is removed. Equivalent is the key word here: the solution set is not changing. The system is just being rewritten in a more useful form.

For example, suppose

$$\begin{cases} 3x+4y=10\\ 2x+y=5 \end{cases}$$

Multiplying the second equation by $4$ would make the $y$ coefficients match. Multiplying the first by $2$ and the second by $3$ would help eliminate $x$. One efficient route is

$$2(3x+4y=10)-3(2x+y=5)$$

which gives

$$5y=5$$

So $y=1$, and substituting into $2x+y=5$ gives $x=2$. The solution is therefore $(2,1)$.

With three equations, repeat the same idea: eliminate one unknown from two pairs of equations, leaving two equations in two unknowns; solve those; then substitute backwards. The working can get bulky, so row reduction gives a tidier notation for the same thinking.

Augmented matrices

A matrix is a rectangular array of numbers arranged in rows and columns. An augmented matrix stores the coefficients of a linear system together with the constant terms, separated by a vertical bar.

For the system

$$\begin{cases} x+3y+z=9\\ 2x+y+4z=21\\ x+5y-3z=-5 \end{cases}$$

the augmented matrix is

$$\left[\begin{array}{ccc|c} 1&3&1&9\\ 2&1&4&21\\ 1&5&-3&-5 \end{array}\right]$$

The columns before the bar correspond, in order, to $x$, $y$ and $z$. The column after the bar contains the constants. This notation saves time because the letters are not rewritten each time, but every row still represents an equation.

Row operations and row echelon form

A row operation is a change to an augmented matrix that preserves the solution set of the corresponding system. The allowed row operations are:

• swap two rows;
• multiply a row by a non-zero constant;
• add or subtract a multiple of one row to or from another row.

Row echelon form is a stepped matrix form where leading non-zero entries move to the right as you go down the rows, with any all-zero rows at the bottom. Reduced row echelon form is a row echelon form in which each leading entry is $1$ and is the only non-zero entry in its column.

For a unique solution in three unknowns, you are aiming for something like

$$\left[\begin{array}{ccc|c} 1&0&0&p\\ 0&1&0&q\\ 0&0&1&r \end{array}\right]$$

where $p$, $q$ and $r$ are constants representing the solved values of $x$, $y$ and $z$ respectively. In that case, the answer is $x=p$, $y=q$, $z=r$.

For the previous example, one row-reduction route is

$$\left[\begin{array}{ccc|c} 1&3&1&9\\ 2&1&4&21\\ 1&5&-3&-5 \end{array}\right] \to \left[\begin{array}{ccc|c} 1&3&1&9\\ 0&-5&2&3\\ 0&2&-4&-14 \end{array}\right]$$

using $R_2\to R_2-2R_1$ and $R_3\to R_3-R_1$, where $R_1$, $R_2$ and $R_3$ are the first, second and third rows of the matrix.

Continuing the reduction gives

$$\left[\begin{array}{ccc|c} 1&0&0&2\\ 0&1&0&1\\ 0&0&1&4 \end{array}\right]$$

So the solution is

$$x=2,\quad y=1,\quad z=4$$

When you write row-reduction by hand, label your row operations beside the matrix. This is not decoration; it makes your algebra traceable and helps you spot where a wrong multiple has crept in.

Recognising no solution or infinitely many solutions

The last row often carries the key row-reduction information. If row reduction produces

$$\left[\begin{array}{ccc|c} 0&0&0&m \end{array}\right]$$

with $m\neq 0$, then the row represents

$$0x+0y+0z=m$$

which is impossible. The system is inconsistent and has no solution.

If row reduction produces an all-zero row,

$$\left[\begin{array}{ccc|c} 0&0&0&0 \end{array}\right]$$

then that row adds no new restriction. The system may have infinitely many solutions, provided there is at least one free variable. A free variable is an unknown that is not fixed by a leading entry and can be chosen as a parameter.

For instance, after reduction you might have

$$\left[\begin{array}{ccc|c} 1&3&1&9\\ 0&1&2&3\\ 0&0&0&0 \end{array}\right]$$

The last row is harmless, but there are only two independent equations for three unknowns. Let $z=t$, where $t$ is a real parameter. Then the second row gives

$$y+2z=3$$

so

$$y=3-2t$$

The first row gives

$$x+3y+z=9$$

so

$$x=9-3(3-2t)-t=5t$$

The infinitely many solutions can be written as

$$(x,y,z)=(5t,3-2t,t),\quad t\in\mathbb{R}$$

This is the algebraic version of saying the planes share a line.

Final augmented-matrix row patterns and the system type they indicate.

Parameters in systems

A parameter is a symbol that represents a fixed but unspecified constant. Parameters often appear in systems of equations when you are asked when a system is consistent, or when it changes from one type of solution to another.

Suppose row reduction leads to the final row

$$\left[\begin{array}{ccc|c} 0&0&0&\lambda-39 \end{array}\right]$$

where $\lambda$ is a parameter. This row represents

$$0x+0y+0z=\lambda-39$$

For the system to be consistent, the right-hand side must be zero, so

$$\lambda-39=0$$

and hence

$$\lambda=39$$

If $\lambda\neq 39$, the last row says $0$ equals a non-zero number, so there is no solution. If $\lambda=39$, the last row becomes $0=0$, and the remaining rows determine whether there is a unique solution or infinitely many solutions.

Technology methods

The syllabus expects both algebraic and technological methods. On a technology-allowed paper, a calculator or graphing tool can solve simultaneous equations directly, or reduce an augmented matrix to row echelon or reduced row echelon form.

The technological method is not different mathematics. It is the same system shown in another form. For a two-variable system, graphing the two lines and finding their intersection is also a valid technological approach. For three variables, matrix or simultaneous-equation tools are usually the sensible route.

Build the habit of linking the calculator output back to the system type:

• output like $(2,1,4)$ means a unique solution;
• a contradictory row such as $[0\ 0\ 0\mid 5]$ means no solution;
• a row of zeros with a free variable means infinitely many solutions.

That final interpretation is where many marks live. The calculator may reduce the matrix, but you still have to say what the result means.