Classification of equations

Summary

• The expression $a+bx$ is said to be a linear expression in the $x$ -variable If $a,b$ are constants and $x$ is a variable.
• An equation is said to be linear if all sides of the equation are linear in the variable.
• The expression $ax^2+bx+c$ is said to be a quadratic expression in the $x$ -variable if $a,b,c$ are constants, $a≠0$ and $x$ is a variable.
• The equation $ax^2+bx+c=0$ is called a quadratic equation if $a, b, c$ are constants, $a≠0$ and $x$ is a variable.
• The expression $x^2+px+q$ is said to be a reduced quadratic expression in the $x$ -variable If $p,q$ are constants and $x$ is a variable.
• The equation $x^2+px+q=0$ is called a reduced quadratic equation if $p,q$ are constants and $x$ is a variable.
• If $a,b, c$ are constants and $x, y$ are variables then the expression $a+bx+cy$ is said to be a linear expression in all the variables . This definition may be extended to an arbitrary number of variables and constants.
• An equation or a system of equations is said to be linear if all sides of the equation(s) are linear in all the variables.
• Two equations or two systems of equations are said to be equivalent if they have the same solutions. For example, $2x+1=5$ is equivalent to $2x=4$ .