### 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.

\[ x=x+1\]

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.

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