1.16 Systems of linear equations

Many problems involve several unknown quantities linked by several conditions at once. A dietitian choosing amounts of three foods to meet targets for protein, carbohydrate and fat, a company working out the prices of three products from sales totals, and the search for a curve through given points all lead to a set of linear equations that must hold at the same time. Solving such a system means finding every combination of values that satisfies all the equations together.

Systems of linear equations have a clear structure. A system can have exactly one solution, infinitely many, or none, and a method called row reduction finds which case applies and what the solutions are. In three dimensions each equation describes a plane, so the three cases correspond to different ways in which three planes can sit in space.

Linear equations and systems

A linear equation is one in which each unknown appears only to the first power, multiplied by a constant. There are no products of unknowns, no powers, roots or functions of them. For example, $2x - y + 3z = 7$ is linear, but $xy + z = 4$, $x^2 + y = 1$

A system of linear equations is often called a set of simultaneous equations. Every system falls into exactly one of three cases: a unique solution, infinitely many solutions, or no solution. A consistent system is one in either of the first two cases. It is impossible for a linear system to have exactly two solutions, or any other finite number greater than one: once there are two solutions, every point on the line through them is also a solution.

Two equations in two unknowns

The three cases as lines

A linear equation in $x$ and $y$, such as $2x - y = 1$, has a straight line as its graph. A solution of two such equations is a point that lies on both lines. The three cases correspond to the three ways two lines in a plane can be arranged.

Elimination

Two equations in two unknowns can be solved by substitution or by elimination. In elimination, each equation is multiplied by a suitable constant so that adding or subtracting them removes one unknown. The same idea, applied repeatedly, solves larger systems.

Three equations in three unknowns

Row operations

The method for three equations generalises elimination. It rewrites the system step by step, using operations that do not change its set of solutions, until the unknowns can be read off one at a time. This process is called row reduction (or Gaussian elimination). Three operations are allowed.

1. Swap two equations.

2. Multiply an equation by a non-zero constant.

3. Replace an equation by itself plus a multiple of another equation.

None of these operations changes the solutions. Any values that satisfy the old equations also satisfy the new ones, since the new equations are built from the old ones; and each operation can be undone by another operation of the same kind, so no solutions are gained either. This is why the final, simpler system has exactly the same solutions as the original. The notation $R_2 \to R_2 - 2R_1$

Triangular form and back substitution

The aim of row reduction is to reach triangular form, in which the second equation contains no $x$ and the third contains neither $x$ nor $y$:

The last equation gives $z$; substituting it into the second gives $y$; substituting both into the first gives $x$. Solving from the bottom up in this way is called back substitution.

The strategy is always the same: use the first equation to eliminate $x$ from the second and third, then use the new second equation to eliminate $y$ from the third.

Setting out with an array

Writing $x$, $y$ and $z$ in every line is slow. Since only the coefficients change, the system can be written as an array of numbers, one row per equation, with a vertical bar separating the coefficients from the right-hand sides. Each row is still an equation: the row $\left(\begin{array}{rrr|r} 0 & 5 & -5 & -20 \end{array}\right)$

Missing unknowns and swapping equations

When the first equation has no $x$, it cannot be used to eliminate $x$ from the others. Swapping it with an equation that does contain $x$ fixes this. Equations with a missing unknown are welcome later in the process, since they are already partly reduced.

Using a GDC

A graphic display calculator (GDC) has a simultaneous equation solver: enter the number of equations and unknowns, then the coefficients and right-hand side of each equation. It returns the solution when the solution is unique, and this is the quickest method whenever a GDC is allowed and the numbers are awkward. When the system has infinitely many solutions or none, the GDC typically shows a message rather than a solution, and some models do not say which of the two cases applies. Row reduction by hand is then needed to identify the case and find the general solution.

A mixture problem gives a typical system with awkward coefficients. The table shows the mass of each nutrient, in grams, in 100 g of three foods, A, B and C (illustrative values).

Systems without a unique solution

Recognising the case during row reduction

Row reduction shows which case a system is in. If the reduced system has a non-zero coefficient of $z$ in the last row, there is a unique solution. If the last row's coefficients all become zero, the right-hand side decides the case.

A row $0 = 0$ means the third equation carries no new information. Only two independent conditions remain on three unknowns, so one unknown can take any value. A row such as $0 = 1$ is a contradiction: no values of $x$, $y$ and $z$ make it true, so the system is inconsistent.

Infinitely many solutions and the general solution

When there are infinitely many solutions, they are described by letting one unknown equal a parameter, usually $\lambda$ or $t$, which can take any real value. The other unknowns are then written in terms of $\lambda$. The result is the general solution; each value of $\lambda$ gives one particular solution.

The general solution can be written in more than one way. Letting $x = \mu$ instead gives $z = \mu + 1$ and $y = 3 - 2\mu$, which describes exactly the same set of solutions: substituting

No solution

If row reduction produces a row of the form $0 = d$ with $d \ne 0$, the equations contradict each other and there is no solution. It is enough to reach this row and state the conclusion; there is nothing to solve.

The two examples differ only in one constant, $13$ or $14$. The left-hand sides determine whether a unique solution is possible; when it is not, the right-hand sides decide between infinitely many solutions and none.

Systems containing a parameter

Questions often include an unknown constant, such as $k$, in the coefficients or on the right-hand side, and ask which values of the constant give each case. The method is to row-reduce with the constant left in, until the last row has the form

Then the coefficient decides whether the solution is unique, and when the coefficient is zero the right-hand side decides between infinitely many solutions and none.

Not every system with a parameter has all three cases. In the next system, the right-hand side of the last row is $0$ for every $k$, so the system is always consistent: it has either a unique solution or infinitely many, but never no solution.

Geometric interpretation: planes

Each equation is a plane

In three dimensions, the set of points $(x, y, z)$ satisfying one linear equation $ax + by + cz = d$ (with $a$, $b$

A solution of a system of three equations is a point that lies on all three planes. The cases of a unique solution, infinitely many solutions and no solution correspond to the ways in which three planes can be arranged.

Consistent systems

When the system is consistent, the three planes have at least one point in common. They may share exactly one point, a whole line, or a whole plane.

When the planes meet in a line, the general solution is a vector equation of the line of intersection. For example, the general solution $x = -1 + \lambda$, $y = 5 - 2\lambda$, $z = \lambda$

the line through $(-1, 5, 0)$ with direction $\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$

If all three equations are multiples of one another, such as $x + y + z = 1$, $2x + 2y + 2z = 2$ and $3x + 3y + 3z = 3$

Inconsistent systems

When the system has no solution, no point lies on all three planes. This can happen in several ways.

The triangular prism is the case that is easy to overlook. Every pair of planes intersects, so a check of pairs suggests nothing is wrong, but the three lines of intersection are parallel and never meet. Row reduction detects it through a row $0 = d$, $d \ne 0$; comparing the normal vectors then shows that no two planes are parallel.

Modelling with systems of linear equations

Fitting a curve through points

A quadratic $y = ax^2 + bx + c$ has three unknown coefficients, so three points on its graph give three linear equations in $a$, $b$ and $c$

Solutions restricted by the context

In a model, the unknowns often stand for quantities that must be non-negative, or whole numbers. A system with infinitely many solutions may then have only a few that make sense. The general solution, together with the restrictions from the context, finds them all.

The general solution describes every mathematical solution, but the model only accepts some of them. Checking a solution against the context (whole numbers, positive amounts, realistic sizes) is part of judging the validity of a model.

Exam-style questions on systems of linear equations

Questions on this content usually combine row reduction with a parameter, a general solution and a geometric interpretation in a single multi-part question on Paper 1, while Paper 2 questions set up a system from a context and solve it with a GDC.

The second question is set in a context and is solved with a GDC. Its data are the following ticket sales at a theatre.

Self-check

• To do:

  I can solve a system of three linear equations by row reduction and back substitution, writing each row operation clearly.

• To do:

  I can solve a system with a unique solution using a GDC.

• To do:

  I can recognise from a reduced system whether there is a unique solution, infinitely many solutions or no solution.

• To do:

  I can write the general solution of a system with infinitely many solutions in terms of a parameter.

• To do:

  I can find the values of a constant that give each type of solution.

• To do:

  I can describe the arrangement of three planes for each type of solution, including the triangular prism.

• To do:

  I can set up a system of linear equations from a context and check that its solutions make sense.