Linear equations

Basic algebra skills: Linear functions

Linear equations

In an equation, two expressions consisting of terms with constants and variables are equated. In this page we will look at the linear equation. We will first look at what a linear equation is and then we will discuss a systematic method of solving linear equations.

Linear equation

An equation with unknown #x# is called linear or first degree if the only terms appearing in it are constants and constant multiples of #x#.

This means that the equation looks like \[a+b\cdot x + c\cdot x+d+\cdots = p+q\cdot x + r + s\cdot x+\cdots\tiny,\] where #a#, #b#, #c#, #d#, #p#, #q#, #r#, #s# are real numbers.

Examples

\[\begin{array}{rcl}x&=&7\\ 2x &=&9\\ 2x+3&=&5\\ 2x+3&=&7x+5\\ 2x+3+x+9 &=& 7x\\ 3x+2&=&0\\ 3&=&0\\ 0&=&0\end{array}\]

The first example is not only an equation, but also the solution of the equation: the only value of #x# that satisfies the equation is #7#.

In general, the equation can be rewritten so that the left-hand side is a linear function of #x# and the right-hand side is #0#.

The last two examples are exceptions. These are equations without terms containing #x#. We also call them linear equations for convenience, although there is no linear function of #x# at play here.

The equation #3=0# with unknown #x# has no solution: the equality is not satisfied for any value of #x#.
For the equation #0=0# with unknown #x#, every value of #x# is a solution. This condition is always met.

Strictly speaking, a linear function in #x# has the form #a\cdot x+b# with #a\ne0#. We have not included in the definition of a linear function the case where #a=0#, i.e. the equations in which, after simplification, #x# no longer occurs, but we have included it in the definition of a linear equation.

Linear equations can be solved using the following method.

Reduction

Linear equations with one unknown can always be solved by reduction, that is, simplifying the equation step by step, in which the same operation is performed on both the left-hand side and the right-hand side.

Subtracting the same term on both sides is such an operation. For example, #2x+3=5# can be reduced to #2x=2# by subtracting #3# from both sides.
Dividing both sides by the same constant, that is unequal to #0#, is also one. For example, #2x =9# can be reduced to the solution #x=\dfrac{9}{2}# by dividing both sides by #2#.
Similar terms can always be grouped together. For example, #2x+3+x+9 =7x# can be simplified to #3x+12 =7x#.

We perform these operations with the intention that, starting from an equation such as #2x+3+x+9 = 7x#, to arrive at an equation of the form #x=3#, because then we will have determined the solution, namely that #x# has the value of #3#.

We follow the steps in this example:

\[\begin{array}{rcl}2x+3+x+9 &=& 7x\\ &&\phantom{xxx}\color{blue}{\text{the equation}}\\ 3x+12 &=& 7x\\ &&\phantom{xxx}\color{blue}{\text{simplified left-hand side}}\\ -4x+12 &=& 0\\ &&\phantom{xxx}\color{blue}{\text{subtracted }7x\text{ from both sides}}\\ -4x&=& -12\\ &&\phantom{xxx}\color{blue}{\text{subtracted }12\text{ from both sides}}\\ x &=& 3\\ &&\phantom{xxx}\color{blue}{\text{divided both sides by }-4}\end{array}\]

The solution is therefore #x=3#.

In other words, a linear equation with unknown #x# can be solved by moving all terms with #x# to the left, moving all constants to the right, combining like terms, and dividing both sides by the coefficient of #x#.

If all terms containing #x# are combined and the coefficient of #x# is equal to #0#, then #x# has disappeared from the equation. What happens then is described later in the General Solution theory.

The idea behind reduction is that we know in advance that the solutions of the original equation are the same as those of the new equation. We then say that the original equation and the reduced equation are equivalent.

Equivalence of equations

Two equations are called equivalent if they have exactly the same solutions.

We also say that one equation is equivalent to the other.

The steps in the above conversion always produce an equation that is equivalent to the previous one.

The equation #2x+3+x+9 = 7x# is equivalent to the solution #x=3#. Each step in the above reduction produces an equation that is equivalent to the original equation.

Each of the two equations #x^6=0# and #x^2=0# is equivalent to #x=0#.

The equations #y^6=-1# and #\frac{1}{y}=0# in the unknown #y# are equivalent: they both have no solution.

Below are some examples of how linear equations can be solved. In the last example, a simple system of linear equations is solved using the same reduction technique.

Find the unique value of #x# for which #6\cdot x+6=6# is true.

Give your answer in the form #x=\ldots# and simplify as much as possible.

#x=0#

#\begin{array}{rcl}
6\cdot x+6&=&6\\
&&\phantom{xxx}\blue{\text{the original equation}}\\
6\cdot x&=&0\\
&&\phantom{xxx}\blue{\text{subtracted }6\text{ from both sides}}\\
x&=&\dfrac{0}{6}\\
&&\phantom{xxx}\blue{\text{both sides divided by }6\text{ }}\\
x&=&\displaystyle 0\\ &&\phantom{xxx}\blue{\text{right-hand side simplified }}
\end{array}#

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