Introduction to Equations

An equation containing a variable is neither true nor false until a particular number is substituted for the variable. If a true statement results from such a substitutions, we say that the substitution satisfies the equation. For instance, the substitution x = 3 satisfies the equation {x^2} = 9, but the substitution x = 4 does not.

An equation that is satisfied by every substitution for which both sides are defined is called an identity. For instance {\left( {x + 1} \right)^2} = {x^2} + 2x + 1is an identity, as is {\left( {\sqrt x } \right)^2} = x. An equation that is not an identity is called a conditional equation. For instance, 2x = 6 is a conditional equation because there is at least one substitution (say, x = 4) that produces a false statement.

If the substitution x = a satisfies an equation, we say that the number a is a solution or a root of the equation. Thus, 3 is a root of the equation 2x = 6, but 4 is not. Two equations are said to be equivalent if they have exactly the same roots. Thus, the equation 2x - 6 = 0 is equivalent to the equation 2x = 6 because both equations have one and the same root, namely, x = 3.

You can change an equation into an equivalent one by performing any of the following operations:

  • Add or subtract the same quantity on both sides of the equation.
  • Multiply or divided both sides of the equation by the same nonzero quantity.
  • Simplify one or both sides of the equation.
  • Interchange the two sides of the equation.

To solve an equation means to find all of its roots. The usual method for solving an equation is to write a sequence of equations, starting with the given one, in which each equation is equivalent to the previous one, but “simpler” in some sense. The last equation should either express the solution directly, or be so simple that its solution is obvious. For example, to solve the equation

2x - 6 = 0

We begin by adding 6 to both sides to get the equivalent equation

2x = 6

Then we divide both sides by 2 to produce the equivalent equation

x = 3

The last equation shows that the root is 3.

Variables representing quantities whose value or values we wish to find by solving equations are called unknowns. A common practice is to use letters towards the end of the alphabet for unknowns, and letters towards the beginning of the alphabet for constants whose values we can assign at will. In particular, the letter x is often used for an “unknown quantity,” and the letters a, b and c are used for constants. A literal or general equation is an equation containing, in addition to one or more unknowns, at least one letter that stands for a constant. For instance,

ax + b = 0

Is a literal equation in which xis the unknown and the constant coefficientsa and bcan be assigned whatever values we please. If we let a = 2 andb = - 6, we obtain

2x - 6 = 0

Whose solution is x = 3.
In applied mathematics, we cannot always follow the convention that unknowns are represented by letters towards the end of the alphabet, because certain symbol reserved for special quantities. For instance, in physics,c is used for the speed of light, m is used for mass, v and is used for velocity, and so on. We shall specify letters represent unknowns to be solved for whenever it isn’t clear from the constant.
            An equation such as 7{x^3} + 3{x^2} + x + 1 = 2x - 5, in which both sides are polynomials in the unknown, is called a polynomial equation. By subtracting the polynomial on the right from both sides of the polynomial equation, we obtain an equivalent polynomial in standard form with zero on the right side.
            7{x^3} + 3{x^2} + x + 1 = 2x - 5
            7{x^3} + 3{x^2} + x + 1 - \left( {2x - 5} \right) = 0   (We subtracted 2x - 5 from both sides)
            7{x^3} + 3{x^2} - x + 6 = 0                    (We combined like terms)
The last equation is in standard form. The degree of a polynomial equation is defined as the degree of the polynomial on the left side when the equation is in standard form. For instance 7{x^3} + 3{x^2} + x + 1 =  2x - 5is a third-degree polynomial equation because, after it is written in standard form, 7{x^3} + 3{x^2} - x + 6 =  0, the polynomial on the left side has degree 3.