Algebraic Equations

An algebraic equation is an expression of the form  e = e'  where e and e' are algebraic expressions.  In an equation e = e' , the algebraic expression e is called the left-side of the equation and e' is called the right-side of the equation.
  Examples
 
  1.     3x -1 = 5x -5

    2.  
  2.      0 =  2x2 +5x -12

    3.  
  3.     A = r2

    4.  
  4.     nm = mn

    5.  
  5.      y =  3x2 +2x -3

    6.  
  6.     n(p + q) =  np + nq

    7.  
  7.     x  =  (-b  (b2 -4ac))/2a

    8.  
  8.    S = at2/2

    9.  
  9.    x2 + y2  = 25
Equations  are used for many purposes. Here are some examples:
  1. To represent a relationship which we can  solve to find  the numerical values of the variables for which the two sides of the equation are equal.   The values which solve the equation are called solutions.   The first, second ,fifth and ninth examples are of this form:
    1. The first equation  3x -1 = 5x -5, has a unique solution,  x = 2,  that is, 2 is the only number which, when uniformly substituted for x in the equation gives us a result  making the two sides equal  32 - 1  = 4 =  52 -5
    2. The second equation,  0 =  2x2 +5x -12, has two solutions,  x = -4  and x = 3/2.
    3. The fifth equation,  y =  3x2 +2x -3, has two variables, x and y,  has an infinite number of solutions. If we solve the equation in the number system of the real numbers then each solution is a pair of numbers <r, r'> with the property that r' = 3r2 + 2r -3.  For example the set of solutions would include the pairs <1.0, 1.0> , <2.0, 9.0>  and <3.3, 25.38>  since

      4.         1.0 =  2(1.0)2 + 21.0 - 3
              9.0  =  2(2.0)2 + 22.0 - 3
              25.38  =  2(3.3)2 + 23.3 - 3

      Use the calculator  to compute some more solutions.   See polynomial functions  and graphing of equations.

    4. The ninth equation, x2 + y2  = 25, also has an infinite number of solutions,  if we graph these solutions we get a circle of radius 5.

      5.  
  2. To represent a relationship between various variables and constants that holds for all choices of  numerical values for the variables  The fourth and sixth equations are of this form
    1. The fourth equation,  nm = mn,  can be interpreted as that statement that for all choices of numbers n and m , multiplying n by m  (that is, computing nm) gives the same result as multiplying m by n (that is, computing mn).  This rule is called the commutative law for addition.
    2. The sixth equation, n(p + q) =  np + nq, expresses the distributive law for multiplication over addition.
    This view of an equation amounts to reading an equation as an assertion that every choice of numerical values for the variables is a solution for the equation. One way to read such an equation, e=e', is as "Every choice of values of the variables (from the numbers system under consideration)  is a solution".


  3. To present a formula which may be used to find the value of a designated variable in terms of the other variables and some constants. Or, to put it another way,  a formula describes a function. The third,  seventh and eighth equations are of this form.
    1. The third equation,  A = r2, expresses the familiar relationship between the area and the radius of a circle using the number = 3.1415...
    2. The seventh equation, x  =  (-b  (b2 -4ac))/2a,  is actually two equations because the symbol , (read as, "plus or minus") denotes both addition and subtraction, thus the two equations are
        3.  x  =  (-b + (b2 -4ac))/2a and x  =  (-b - (b2 -4ac))/2a
      These equations express the relationship between  the solutions of the equation   ax2 +bx + c = 0  (or the equation 0 = ax2 +bx + c)  and the values of the coefficients a, b and c. For example,  in our second equation, 0 =  2x2 +5x -12, the coeficients are  a = 2, b =5 and c = -12.  Substituting these values into the equation we get   x  =  (-5 (52 -42(-12)))/22  =  (-5 (25+96))/4 = (-5 121)/4  =  (-5  11)/4 so, interpreting as addition we get x = (-5 + 11) = 6/4  = 3/2,  and interpreting as subtraction we get x = (-5 -11)/4  = -16/4  =   -4,  exactly the solutions given earlier. This (pair of ) equation(s), is called the quadratic formula.   We will see more the quadratic formula  later.
    3. The eighth equation, S = at2/2,  expresses the distance, S,  an object will go in time, t, when  accellerating at rate a. For example, a falling body (near the surface of the Earth)  is accellerated at the rate of approximately 32 feet per second each second.
    For an equation to be a formula in this sense it is necessary that "one side"  of the equation (generally the left-side) consist of a single variable that does not occur on the other side of the equation.   In this case, given any assignment of values to the variables on the right-side we can evaluate the right-side and assign the resulting value to the variable on the left-side to get a solution to the equation.