The notion of linear equation

Systems of linear equations and matrices: Linear equations

The notion of linear equation

Suppose that \(x\) represents the number \(3\) and \(y\) the number \(2\). Then, for instance, the following holds: \(x+1=6-y\). This means that \(x=3, y=2\) satisfies the equation \(x+1=6-y\). There are many more equations that are satisfied by the numbers #2# and #3#.

In practice, the situation is the other way around: \(x\) and \(y\) are unknown numbers that satisfy the equation \(x+1=6-y\) and we are after the possible values of \(x\) and \(y\). In other words, we want to solve the equation. This can be done by reduction, i.e., by iteratively writing an equation that is simpler than the previous one and still has the same solution. In the example chosen, the equation can be reduced to \(y=5-x\), which means that for a randomly chosen value for \(x\), say \(x=a\), the value of \(y\) is given by \(y=5-a\).

However, the example chosen is a special type: it is a linear equation in \(x\) and \(y\). In this section we focus on the case of one linear equation in one or more unknowns.

Linear equation

Let \(x_1,\ldots, x_n\) be variables.

A linear equation with unknowns \(x_1,\ldots, x_n\) is an equation that can be reduced, by use of elementary operations, to a (linear) basic form \[a_1x_1 + \cdots + a_nx_n + b = 0\] where \(a_1,\ldots,a_n\) and \(b\) are real or complex numbers. We also speak of a linear equation in \(x_1,\ldots ,x_n\).

An elementary operation is understood to be the expanding of brackets, regrouping of parts of expressions, adding or subtracting equivalent expressions on both sides of the equation, or multiplying and dividing on both sides of the equation by a number distinct from zero. We speak of an elementary reduction if all the steps within the reduction are elementary operations.

The expression to the left of the equal sign ( \(=\) ) is called the left-hand member or the left-hand side of the equation (above, this is \(a_1x_1 + \cdots + a_nx_n + b\) ), and the expression to the right is called the right-hand member or the right-hand side (above, this is \(0\)).

The expressions \(a_1x_1, \ldots, a_nx_n\) and \(b\) in the left-hand member of the basic form are called terms. For each index \(i\), we call the number \(a_i\) the coefficient of \(x_i\). Terms that do not contain an unknown are called constant terms, or constants for short (above, these are the numbers \(b\) and \(0\)).

A list \(\rv{s_1,\ldots, s_n}\) of \(n\) numbers is called a solution of the equation if entering \(x_1=s_1, \ldots, x_n=s_n\) turns the equation into a true statement. All values of \(x_1,\ldots ,x_n\) in which the equation is true constitute the solution of the equation.

Two linear equations are called equivalent when they can be transformed into one another by elementary reduction.

If an equation can be reduced to another by elementary transformations, then the two equations are equivalent.

Substituting \(n=1\), \(a_1=2\), and \(b=3\) in the above basic form of a linear equation gives \(2x_1+3=0\). A solution of this equation is \(x_1=-\frac{3}{2}\). In fact, it is the solution: there are no others.

We then say that \(x_1=-\frac{3}{2}\) is the solution of the equation \(2x_1+3=0\).

The equation \(x_1+2=-(x_1+1)\) is an equivalent linear equation, and thus has the same solution.

There is no unique basic form: the equations \(2x-2=0\) and \(x-1=0\) both have the basic form but are different and can still be carried over into one another by use of elementary operations.

The equations \(x_1+x_2=x_1\cdot x_2\) and \(x_1+\sqrt{x_2}=0\) are not linear, because the product \(x_1\cdot x_2\), respectively, the square root \(\sqrt{x_2}\) is present.

Although the equations \(\frac{x-1}{2x-3}=4\) and \(|x-1|=|2x-3|\) can be solved using linear equations (check this yourself), we still cannot call them linear equations, as the left-hand side of the first equation is a rational expression and the second equation contains absolute values. As a result, there is no elementary reduction to the basic form of a linear equation.

The equation \((x-1)^2=(x+1)^2\) is considered linear, because expanding the brackets and regrouping terms lead to the equation \(4x=0\). It would be too artificial to consider this equation as a quadratic equation \(0x^2+4x=0\), just because there is a square in the original equation.

Conversely, we call the equation \(\sqrt{(x-1)^2+4x}=0\) not linear, even though it can be reduced to \(x+1=0\). However, the reduction does contain a non-elementary step: the transition from \(\sqrt{(x+1)^2}=0\) to \(x+1=0\).

The equivalence of equations concept may also be used for non-linear equations:

Two equations are called equivalent if they have the same solution.

Each of the two non-linear equations \((x-1)^3=0\) and \((x-1)^2=0\) is equivalent to the linear equation \(x-1=0\). However, the first two equations are non-linear because, in order to reduce to the linear equation \(x-1=0\), a non-basic simplification is needed.

Regarding the format of linear equations, there are conventions that we commit to as much as possible. For example, if the number of variables in a linear equation is small, then usually different letters are used instead of indexed names \(x_1, x_2,\ldots\). In this vein, we write \(2x+3y+4=0\) instead of \(2x_1+3x_2+4=0\).

In this chapter we will use the two formats together. However, in computer exercises we will favour letters for variables, because they are easier to implement than indexed names.

Non-elementary operation

Squaring both sides of an equation is an example of an operation that is not elementary and that does not always transform an equation into an equivalent one. For instance, squaring both sides of the equation #x=5# gives #x^2=25#, an equation having #x=-5# as a solution next to #x=5#. Besides, this operation transforms many linear equations into non-linear ones.

Notation

Often, instead of \(x\), \(y\), and \(z\), names like \(x_1\), \(x_2\), and \(x_3\) are used to indicate the unknowns. For example, \(x_1+4x_2+5x_3+6=0\) instead of \(x+4y+5z+6=0\). This makes it easier to write the theory, methods, and techniques for an arbitrary number of \(n\) unknowns.

Is the equation in #x# given below linear?

\[ x=x+1\]

Yes

The equation can be reduced to #0\cdot x-1=0 # and is therefore linear according to the definition.

It is a degenerate situation where the unknown is not really present in the equation.

Incidentally, the equation has no solutions.

New example