The notion of linear equation in one unknown

Solving equations and inequalities: Linear equations in one unknown

The notion of linear equation in one unknown

Solving a linear equation in one unknown by reduction Suppose that \(x\) symbolises the number \(3\), then we have for example \(x+1=7-x\). This means that \(x=3\) satisfies the equation \(x+1=7-x\). For the number 3 one can write many more equations that are satisfied.

In practice, the situation is the reverse: then \(x\) is an unknown number satisfying the equation \(x+1=7-x\) and you are curious about the value of \(x\). In other words, you want to solve the equation. This can be done by reduction, i.e. by writing an equivalent equation that is simpler than the previous one but has the same solution, and continuing this rewriting process until you arrive at the form \(x=\ldots\). In the example chosen, it may go as follows:

Solving the equation \(x+1=7-x\).

Add to the left- and right-hand side \(x\) by: \(\; x+1+x=7-x+x\),
which simplifies to \(\;2x+1=7\).
Subtract \(1\) from the left- and right-hand side: \(\; 2x+1-1=7-1\),
which simplifies to \(\;2x=6\).
Divide the left- and right-hand size by \(2\): \(\;2x\div 2=6\div 2\),
which simplifies to \(\;x=3\).

So, the solution is \(x=3\).

The steps in solving by reduction are:

adding or subtracting the same term on both sides of the equation;
multiplying or dividing both sides by a nonzero number;
collecting similar terms.

Look at more examples of solving what is called a first degree equation with one unknown or a linear equation with one unknown :

Determine the exact solution \(x\) of the equation \[-7x-6=-55\] by the reduction method. This means that you go in small steps to the solution by entering each time an equivalent equation that is simpler and has the same solution as the equation at the start.

The reduction goes in a few steps from the starting point \(-7x-6=-55\):

1. Add 6 to the left- and right-hand side: \(\quad-7 x -6 +6=-55+6\quad\) or \(\quad-7 x=-49\)

2. Divide the left- and right-hand side by -7: \(\quad x=\frac{-49}{-7}\quad\) or \(\quad x=7 \)

Conclusion: the exact solution of the equation \(\quad -7x-6=-55\quad\) is \(\quad x=7\).

The above reduction can also be written in less words and with the implication arrow \(\implies{}\!\!\!\), possibly with a label that indicates what operation is applied to the equation:
\[\begin{aligned}-7x-6=-55 &\stackrel{\blue{+6}}{\implies} -7x -6 +6=-55+6\\ \\ &\implies -7x=-49 \\ \\ &\stackrel{\blue{{}\div(-7)}}{\implies}\frac{-7 x}{-7}=\frac{-49}{-7}\\ \\ &\implies x=7\end{aligned}\]

New example

We end this theory page with common terminology that we will use henceforth.

General terminology Let \(x\) be a variable.

A linear equation with unknown \(x\) is an equation that can be reduced, by elementary operations, to a (linear) basic form \[ax+ b = 0\] where \(a\) and \(b\) numbers. We also speak of a linear equation with unknown \(x\) and a linear equation in \(x\).

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

An elementary operation is understood to be expansion of brackets, regrouping of subexpressions, addition or subtraction of equivalent expressions on both sides of the equation, or multiplication or division 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.

An equation consists of two parts: The expression to the left of the equal sign (\(=\)) is called the left-hand side or the left side of the equation (above, this is \(ax + b\) ), and the expression to the right is called the right-hand side or the right side (above, this is \(0\)).

The expression \(ax\) and \(b\) in the left-hand side of the basic form are called terms. The number \(a\) is the coefficient of \(x\). Terms that do not contain an unknown are called constant terms, or constants for short (above, these are the numbers \(b\) and \(0\)).

A number \(s\) is called a solution of the equation if entering \(x=s\) turns the equation into a true statement. All values of \(x\) for which the equation is true constitute the solution of the equation.

Two linear equations are called equivalent when they have the same solutions because they can be transformed into one another by elementary reduction.
To indicate that two equations are equivalent, the symbol \(\Leftrightarrow\) can be used; for example \(4x=2\Leftrightarrow 2x=1\) and \(2x=1\Leftrightarrow x=\tfrac{1}{2}\).

If two equations can be reduced to the same basic form, then the two equations are equivalent.

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 equations \((x-1)^2=(x+1)^2\) and \((x+2)^2=(x+1)^2\) are considered linear, because expanding the brackets and regrouping terms lead to the equations \(4x=0\) and \(2x+3=0\). It would be too artificial to consider these equations as quadratic equations \(0x^2+4x=0\) and \(0x^2 +2x+3=0\) on to grasp, 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 concept of equivalence of equations may also be used for non-linear equations:

Two equations are equivalent if they have the same solution(s).

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-elementary simplification is needed. They are examples of third-degree and second-degree equation

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.