Linear

The algebraic operations of a number system are the operations of addition, subtraction, multiplication, division, and the extraction of roots. For a proper consideration of root extraction we must extend our number field to the complex. For now, however, we will be concerned only with the real numbers.

In this section we will be dealing with equations of the form:

ax + b = 0 [a, b Real and "a" is nonzero]

ax + b = 0

  • ax + b - b = 0 - b [subtract the value "b"]

    Whatever we do to one side of an equation must likewise be done to the other. Therefore:

  • ax = -b

  • ax/a = -b/a [divide through by the value "a"]

  • x = -b/a [our only solution.]

    • TWO STEP STRATEGY FOR SOLVING ALGEBRAIC EQUATIONS

    1) Simplify and Condense like terms on each side of the sign;

    2) "ISOLATE" the variable(s)

    A function is a relation between two sets of elements called, respectively, the domain and the range such that a rule of association is defined mapping an element of the domain to one and only one element of the range. If we use "f" to designate this function, it is customary to use "f(x)" to designate the element corresponding to "x," and this value, "f(x)," is called the image of "x." The elements of the domain may be considered as the input, the elements of the range, the output.

    If "f" is a function defined on R ["R" stands for Real Numbers] by

    f(x) = ax + b, with a, b real,

    then the graph of "f" is a straight line, and for this reason "f" is said to be a linear function. The notation, "f(x)" can be thought of as standing for or representing ax + b in a more succinct way, like any other symbol that stands for somthing else.

    Any equation in one unknown of degree one can be put into the above form through suitable simplifications. These simplifications are exactly equivalent to what the reader has become accustomed to in his/her study of Arithmetic.

    Level One

  • Grouping like terms:

    "Four ducks plus six geese plus three cows plus two ducks plus one goose" can be simplified to, "six ducks plus seven geese plus three cows." I am not trying to be purposely condescending but, one of the major difficulties people have with algebra, and mathematics in general, is reading more difficulty into it than there really is. That is to say:

    "6x + 3y + 9z + 5x +4z + 10y = 11x + 13y +13z

    x = duck
    y = goose
    z = cow

    These letters stand for things, number values most generally; and they are called variables because the values they stand for can vary.

    Admittedly this is the simplest case, however, check it out...

  • Simplifying and Grouping like terms where fractions are involved

    "1/2 + 1/3 + 1/12 = 6/12 + 4/12 + 1/12 = 11/12"

    bears a striking resemblance to:

    x/2 + x/3 +x/12 = 6x/12 + 4x/12 + x/12 = 11x/12

    You no doubt learned how to find the common denominator in Arithmetic; it's no different in Algebra; don't let the addition of letters buffalo you; just deal with them as an attachment to whatever number of them there are. Next case...

  • Simplifying and Grouping like terms of a more complex nature

    "3/4/5/6 + 2/3/7/8 = 3/4 × 6/5 + 2/3 × 8/7 = 18/20 + 16/21"

    Here we do the inversion trick which is nothing more than multiplication by a form of the number one. This is a crucial idea used throughout algebra.

    --- forms of one ---

    anything other than zero or infinity over itself is one or equals one.

    3/4/5/6 can be simplified by

    3/4/5/6 × 6/5/6/5 [form of one].

    The denominators cancel to "1" and we're left with 3/4 × 6/5 which is the same
    as if we had merely inverted our original fractional denominator and multiplied.
    To continue...

    3x/4/5/6x = 3x/4 × 6x/5 = 18x²/20

    You'll notice the term x² here. This is notation for x × x; it is nothing more than:

    4 × 4 = 4² = 16, for example.

  • Simplifying and Grouping across equal signs

    There is a trick to know when dealing with fractions:

    3x/4 + 6x/10 = 2x/5

    What do we do with this?

    Find the common denominator and simplify the left side:

  • 3x/4 × 10/10 [one] + 6x/10 × 4/4 [one] =
  • 30x/40 + 24x/40 =
  • 54x/40

    Which gives us:

  • 54x/40 = 2x/5

    Now, we can multiply both sides by 5/2x or... when we bring a number over to the other side of the equal sign, we invert it, same difference, only faster.

  • 54x/40 × 5/2x = 1
  • 27x/8 = 1 [cancelling]
  • 27x = 8 [when we bring a number over to the other side...]
  • x = 8/27
    [isolate the "x" by dividing through by 27].

    So far we have only seen single term combinations. But terms with more than one element are treated in a similar manner:

  • Simplifying and Grouping larger terms

  • (x + 2)/(x + 3) + x/(x + 3) = (2x + 2)/5
  • (2x + 2)/(x + 3) = (2x + 2)/5 [same denominator, add numerators]
  • 5/(x + 3) = (2x + 2)/(2x + 2) [when we bring a "number" over to the other side...]

    Here's the key: whatever is within the parentheses ["( )"] is considered a "single number"]

  • 5/(x + 3) = 1 [(2x + 2)/(2x + 2) is a form of one]
  • 5 = (x + 3) [when we bring a number over to the other side...]
  • 5 - 3 = x [whatever we do to one side we must do to the other; isolate the variable]
  • 2 = x

  • Simplifying and Grouping larger terms with different denominators

  • 5/(x + 3) - 10/(x + 2) = 15/(5x + 10)
  • (5(x + 2) - 10(x + 3))/(x + 3)(x + 2) = 15/(5x + 10) [our common denominator for the left side is (x + 2)(x + 3) so we must multiply each numerator by a form of one which will give it to us, that is, 5 × (x + 2)/(x + 2) and, 10 × (x + 3)/(x + 3)]
  • (5x + 10 - 10x -30)/(x + 3)(x + 2) = 15/5(x + 2) [when multiplying through it is important to remember to distribute the negative sign, that is, -10(x + 3) is equal to -10x - 30 or, -(10x + 30) ... same diference. it is always understood that a "1" is out in front of the parenthesis sign, "(". therefore, what we actually have is, -1 × (10x + 30). also, we have factored out a 5 from (5x + 10)]
  • -(5x + 20)/(x + 3)(x + 2) = 3/(x + 2) [combining like terms, and dividing 15 by 5; or, alternatively, we may think of 15 as being composed of 5 × 3 which when placed over 5 as a denominator allows us to cancel as 5/5 is a form of one.]
  • -(5x + 20) = 3(x + 3) [what we have done here is to bring the denominator from the left side over to the right necessitating its inversion, and cancelled one (1) in the form, (x + 2)/(x + 2)]
  • -5x - 20 = 3x + 9 [multiplying through by the understood -1, and 3]
  • -29 = 8x [simplifying the equation. adding and subtracting suitable terms and numbers in order to eliminate and condense, remembering that whatever we do to one side we have to do to the other; or, alternatively, bringing a term or number over to the other side changes its sign]
  • x = -29/8 [isolate the variable by division, or, bringing a factor over to the other side inverts its numerator/denominator position.]

    Quite a bit was thrown at you with this last one. To sum up:

  • factoring out a common term;
  • utilizing the idea of different forms of one (1); that is: a/a where a is a rational number;
  • condensing and simplifying fractions;
  • bringing terms across the equal sign;
  • applying the distributive law and the understood one (1)

    That's all we used. It's worthwhile to point out at this time a significant idea to keep in mind. Whenever we are asked to "solve for the varaible" in a "one unknown" equation, we already know what form the answer is supposed to be in, or rather, what it's supposed to look like. When solving for "x" we know that the last line in the process of gradually zeroing in on the solution is going to be:

    x = something

    And with just a few fundamental rules concerning the grammar or structure of algebra, equivalent to arithmetic, we can arrive at this last line.

    Another significant point can also be made at this time. I have come up against a recurring problem with students concerning the minus or negative sign. It tends to trip some people up, particularly when it comes to solving equations with terms in parentheses demanding the use of the distributive law. This is quite understandable. The reason is this: the minus sign has a dual role, an ambiguity about its function. In fact, the main step in the modern development of Algebra was the evolution of a correct understanding of negative quantities. It really makes no difference whether the minus sign is to be associated with a number or to be regarded as the symbol for subtraction. Even though the sign of a number and the symbol of an operation are entirely different logical notions,

    a - b is defined to be the same as a + (-b)

    what the solver of equations needs to keep in mind in order to avoid confusion is that in the midst of any given equation, once he has decided firmly in his mind just which interpretation works best for the given situation, he has to stick with it! That is to say, no changing horses in the middle of the stream. If you decide one way or the other, then changing the designation will most assuredly effect an incorrect solution. Choosing thusly brings clarity and a sense of confidence and control.

    Example:

    5x - 7
    -(3x + 3)
    ----------
    2x - 4

    Level Two

    Considering the equation of the line,

    ax + by + c = 0
    [a, b, c, -- real numbers; x, y, -- coordinate axes]

    with the "y" value equal to zero produces the linear equation

    ax + c = 0.

    Example

  • 5x + 3 = 0
  • 5x = -3 [five times the unit value for x lies at the position -3 on the real line]
  • x = -3/5 [normalizing, 5x + 3 will lie at the intersection if and only if x has this value]

    From this vantage point we are now looking at a one-dimensional representative model embedded in a two-dimensional euclidean plane or space. If we look at the expression -- ax + c -- as a position on the real number line having the value zero at the intersection of the two axes -- x and y -- we can deduce information from it alone.

    Considering the above example, we want to find the value for "x." What do we know of our position? We know by the legitimacy of the expression that the value "x" must be less than the value "c." Why? If "x" were anything but its correct value, "a" would have to be decreased by an amount equal to (ax+c)/x. So, if "x" were equal to or greater than "c," "a" would, of course, have to be equal to or less than -1.

    Taking five 'steps' equal to 3/5 in the negative ("negative" here simply means "opposite to positive") direction, and then turning about and walking three 'steps' of length equal to "1," we arrive at the intersection, zero.

    So, as the position of "ax" is equal to "-c" (for standard form ax + c), in order to normalize "x" and thereby discover its value we have to factor "ax" by the value "a" which, by the grammar of algebra, demands that we must also factor "-c" by this same value. We arrive at x = -c/a. This confirms the only piece of information we had at the beginning, that is, the expression -- ax + c -- is a map directing us, by a roundabout route, to the intersection of the two axes, the number zero on the real line.

    At the root of mathematics is the ability to create symbols and to discern patterns. Pattern recognition is fundamental, as is language; mathematics is the quintessential language of symbol manipulation. Once ideas and concepts are defined and classified, associations and relationships lending structure can be drawn. In order to eliminate and avoid fuzziness and vague characterizations, a rigorous logic needs to be applied to these symbols for the sake of clarity and definition. These symbols live in a realm all their own, self-consistent on horizontal levels of equivalence, and developmental on vertical lines of increasing complexity. This is the basis for a metaphorical approach to algebra; there are no breaks in the continuity of ideational growth; and what is true at any level will likewise be so at any higher, more general and inclusive one.

    The arithmetic of algebra, its grammar and syntax, if you will, once solidly grounded and internalized, can act as a springboard for this exploration into the world of symbols and the study of their relationships. The combination of arithmetical grammar and sets of equivalence classes precisely defined sums up Algebra at any conceivable level.

    We can thus speak of "The Algebra of..." as in the algebra of exponents and logarithms; the algebra of trigonometric relationships; the algebra of polynomials; the algebra of differential equations; and so forth. The geometries themselves can be reclassified and framed by an algebra that defines their respective invariant properties [Felix Klein]. Even though each language of the world has its own grammar, they are yet translatable one to the other. Thoughts are generic, as is Algebra. There are rules to be learned and analytical methods to be discovered.

    Table of Contents
    Linear Systems || Quadratics