Solving Linear Equations

When asked to "solve" for the missing "x" variable in a linear equation, we are actually finding the first coordinate of an ordered pair where the "y" value has been established. More specifically, when asked to "solve" for x in the equation 25 = 3x + 1, the y value has been assigned the number 25 and we are looking for the x value. This problem could be solved graphically by graphing y = 3x + 1 on a a graphing utility, and consulting the table of values for y = 25. OR...

This problem could be solved using the Real Number Properties (describing how numbers behave) and the Laws of Algebra. You may use one or more of the properties to perform "transformations" on the original algebraic statement until the correct form is achieved.

The more frequently used properties include:

Addition Property: add same quantity to both sides of =
Subtraction Property: subtract same quantity from both sides of =
Multiplication Property: multiply both sides of = by same quantity
Division Property: divide both sides of = by same quantity
Distributive Property: multiply a colleciton of terms by a specified number
Additive Inverse: a + (-a) = 0
Multiplicative Inverse a(1/a) = 1
Additive Identity: a + 0 = a
Multiplicative Identity: a * 1 = a
Substitution Property: replace equal values with each other

* Simplification is used to combine similar terms for reduction of # of terms

Example (1): Solve 12x + 175 = 487

Solution:

          12x + 175 = 487             Given Problem
          487     = 12x + 175         Optional step-to be consistent with Linear discussion
          487-175 = 12x + (175 -175)  Subtraction Property
          312     = 12x + (0)         Substitution
          312     = 12x               Additive Identity
          312/12  = 12x/12            Division Property
          26      = (1) x             Multiplicative Identity
          26      =  x                Simplify




Example (2): Solve  5p - 9 = 3(p - 7)



Solution:        5p - 9   = 3(p - 7)           Given Problem
        5p - 9   = 3p - 21            Distributive Property
     (5p -3p) -9 = (3p -3p) -21       Subtractionn Property
        2p    -9 =    0    - 21       Substitution/Simplify
        2p    -9 =     -21            Additive Identity
    2p (- 9 + 9) =   -21 + 9          Addition Property
      2p + 0     =    -12             Simplify
             2p  =    -12             Additive Identity
           2p/2  =    -12/2           Division Property
           (1)p  =    -6              Multiplicative Identity
              p  =    -6              Simplify

As expertise is gained in solving basic linear equations, some of the above steps may be combined. Each and every step in the solution process must be justified with an accepted Property or Law.

Another form of equation which requires discussion is called "Literal Equation". This type of equation has more than one variable present. You must be advised "which" variable is to be solved for. The procedure is exactly as was demonstrated for other linear equations. The solution for a Literal Equation will not be numerical but rather will be an expressionn in terms of the other variables present.

Example (1): Solve d = r * t : for r

Solution:

   
              d  = r * t              Given Problem
             d/t = r * t /t           Division Property
             d/t = r *  1             Substitution
             d/t = r                  Multiplicative Identity

Note: All formulas containing more than one variable actually represent many formulas if you consider them as Literal Equations.

[TOC] [Mathematical "Models"] [Functions] [Probability] [Direct Variation] [Solving Linear Equations]
[Analyzing Equations of Lines] [One Variable Inequalities] [Arithmetic Sequences] [Geometric Sequences] [Irrational Numbers] [Complex Numbers] [Quadratic Functions] [Solving Quadratic Equations] [Conic Sections] [Variation] [Exponents and Roots] [Solving Radical Equations] [Function Operations] [Polynomial Functions] [Rational Expressions] [Rational Functions] [Solving Rational Equations] [Exponential Functions] [Logarithmic Functions]