Algebra
LINEAR EQUATIONS
The root(s) of an equation (conditional equation) is any value(s) of the literal number(s) in the
equation that makes the equation true. Thus, 1 is the root of the equation 3x + 5 = 8 because
x = 1 makes the equation true. To solve an algebraic equation means to find the root(s) of the
equation.
The application of algebra is practical because many physical problems can be solved using
algebraic equations. For example, pressure is defined as the force that is applied divided by the
area over which it is applied. Using the literal numbers P (to represent the pressure), F (to
represent the force), and A (to represent the area over which the force is applied), this physical
relationship can be written as the algebraic equation
. When the numerical values of the
P
F
A
force, F, and the area, A, are known at a particular time, the pressure, P, can be computed by
solving this algebraic equation. Although this is a straightforward application of an algebraic
equation to the solution of a physical problem, it illustrates the general approach that is used.
Almost all physical problems are solved using this approach.
Types of Algebraic Equations
The letters in algebraic equations are referred to as unknowns. Thus, x is the unknown in the
equation 3x + 5 = 8. Algebraic equations can have any number of unknowns. The name
unknown arises because letters are substituted for the numerical values that are not known in a
problem.
The number of unknowns in a problem determines the number of equations needed to solve for
the numerical values of the unknowns. Problems involving one unknown can be solved with one
equation, problems involving two unknowns require two independent equations, and so on.
The degree of an equation depends on the power of the unknowns. The degree of an algebraic
term is equivalent to the exponent of the unknown. Thus, the term 3x is a first degree term; 3x2
is a second degree term, and 3x3 is a third degree term. The degree of an equation is the same
as the highest degree term. Linear or first degree equations contain no terms higher than first
degree. Thus, 2x + 3 = 9 is a linear equation. Quadratic or second degree equations contain up
to second degree terms, but no higher. Thus, x2 + 3x = 6, is a quadratic equation. Cubic or third
degree equations contain up to third degree terms, but no higher. Thus, 4x3 + 3x = 12 is a cubic
equation.
The degree of an equation determines the number of roots of the equation. Linear equations have
one root, quadratic equations have two roots, and so on. In general, the number of roots of any
equation is the same as the degree of the equation.
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