Linear equations with a single unknown

Functions: Lines and linear functions

Linear equations with a single unknown

Let $x$ be a variable.

A linear equation with unknown $x$ is an equation that has form

$a\cdot x+b=0,$ in which $a$ and $b$ are real numbers.

Solving the equation is finding all values of $x$ for which the equation is true. Such a value is called a solution of the equation. The values of $x$ for which the equation is true, form the solution of the equation, also called the solution set.

Equations with $x$ as unknown are called equations in $x$ .

The expression to the left of the equal sign ( $=$ ) is called the left-hand side of the equation (for the equation above this is $a\cdot x+b$ ), and the expression on the right of it is the right hand side (for the equation above this is $0$ ).

The expressions $a\cdot x$ and $b$ in the left hand side are called terms. Because $b$ and $0$ occur without $x$ , they are called constant terms, or simply constants. The number $a$ is called coefficient of $x$ .

For $a=2$ and $b=3$ the equation is $2x+3 = 0$ , and $x = - \dfrac{3}{2}$ is a solution. It is even, the solution: there are no other. We say that $x= -\dfrac{3}{2}$ is the solution of the equation $2x+3 = 0$ . The solution set can also be specified as $\{-\frac{3}{2}\}$ .

The type of equation $2x+3=5x-6$ is very close to the real linear equation to: by moving all terms to the left hand side and taking them together, we can rewrite it as a real linear equation $-3x+9=0$ . Therefore, this type is also called a linear equation. Even more general: if all terms in the equation are constants or constant multiples of $x$ , then the equation is called linear.

In terms of a function solving the equation is finding all points $x$ where the linear function $a\cdot x+b$ is equal to $0$ .

Instead of linear we can also say of first degree, because the highest degree in which the unknown $x$ occurs is no higher than $1$ . The name of first degree comes from the theory of polynomials.

The words terms and constants have been introduced earlier.

According to the definition, the equations $0=0$ and $0=1$ are also linear. After all, they can be rewritten as $0\cdot x+0 = 0$ and $0\cdot x -1 = 0$ , respectively. On the one hand, we could have prevented this by requiring that $a$ be nonzero in the equation $a\cdot x + b = 0$ . On the other hand, it is convenient to work with the general case in which the coefficient $a$ of $x$ and the constant term $b$ are arbitrary numbers. Therefore, we do not require $a\ne0$ .

In this chapter we will first deal with linear equations with a single unknown and later we will deal with linear equations with two unknowns.

Solve the equation $5 x - 30=0$ with unknown $x$ .

$x = 6$

This answer can be found as follows.

$\begin{array}{rcl} 5 x &=& 30\\ &&\phantom{xxx}\color{blue}{30\text{ added to each side}}\\ x &=&\dfrac{ 30}{5}\\ &&\phantom{xxx}\color{blue}{\text{both sides divided by }5}\\ x &=& 6\\ &&\phantom{xxx}\color{blue}{\text{simplified}} \end{array}$

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