Algebra

[last updated: 2021-04-08]
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Algebra is easy once you know the tricks.
Some Basics:
- The First Trick:
  Algebra uses letters instead of numbers:
  - x, or y, or z, or E, or I - these letters all represent numbers.
    These letters represent numbers that are values or measurements of some real world parameter.
  - So x and y might be coordinates on a graph, describing a position in space.
    E and I might be volts and amps in an electronic circuit.
  - The letters are often said to represent variables.
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- Trick #2:
  Algebra uses Equations:
  Equations are also sometimes called "formulas" or "expressions".
  - Equations are things that have an equal sign:
    The total of everything on the left side of the equal sign is equal to (is the same number as)
    the total of everything on the right side of the equal sign.
    
    Example equations:
    These 3 equations are 3 different ways that can be used to say the same thing:
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- Trick #3:
  Rules of Equality:
  - These are kind of obvious and intuitive, but they're important, so we'll be explicit:
  - If: a = b
    then: b = a
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    - Trick #3a:
      Replace or "Plug in" an equal value for a variable:
    - If: a = b
      and: b = c
      
      then: you can take the c from the second equation, where b = c,
      and you can "plug in" the c into the first equation in place of the b.
      
      This results in: a = c
    - This is an important rule - you will use it often to solve complicated problems.
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A Little Deeper:
Algebra is used to solve Story Problems:
- These are stories, that is, words that describe some situation, that have numbers in them,
  and we're asked to find some other number - the unknown value or variable.
- That is, we're asked to Find an Equation
  that will calculate the unknown that we want.
- Trick #4:
  to Solve Story Problems: 1: Start with What you Want:
  You Want to Find the Unknown Variable.
  More precisely: You Want an EQUATION that will Calculate the unknown variable.
  More specifically: You want an Equation that has the unknown variable on the left side of the equals sign, alone, by itself,
  and nothing but known variables on the right side of the equals sign.
  
  First decide the letter you will use to represent the unknown variable.
  - Here's a simple story problem:
  - So what do you want?
    "How much current flows..."
  - The current flow is the unknown variable.
    Let's choose to use the letter I to represent the current flow.
    This is arbitrary. We could choose any letter. However "I" is often used to represent current, so we'll stick with that common convention.
  - So now we know what we want:
    we want an equation that has our unknown I on the left side,
    and whatever else on the right side:
    This will let us calculate the value of our unknown variable I, and solve the problem.
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- Trick #5:
  to Solve Story Problems: 2: List What you Have - what you're Given:
  Decide the letters you will use to represent the known variables.
  - In the problem above, the known variables, the numbers that we're Given are:
    15 vdc, and 1000 ohms
    
    These are the known variables.
  - Let's decide to use E and R to represent the vdc and ohms.
    This gives us:
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- Trick #6:
  to Solve Story Problems: 3: Find Whatever Else You Need:
  - This is where it gets squishy
    because what you need for one problem will not be the same as what you need to solve a different problem.
  - Start by looking at what you have: E & R
    Now look at what you want: I
    
    Now search your memory or notes to find some formula/equation/rule that has all those things.
  - This example is an easy problem to solve for someone with knowledge of basic electronics,
    because we know Ohm's Law: E = I * R has everything we need.
  - Now we have everything we need to solve the problem:
    Now we just need to convert what we have
    into what we want: I = ...
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Even Deeper:
Juggling:
- Juggling is what you do to move things around in your equations so they are in the form you want.
  In the example, what we HAVE is: E = I * R.
  We see that I, the unknown variable we're trying to solve for, is on the right side of the equation.
  But what we WANT, is I, by itself, on the Left side of the equation.
  So we juggle things around.
- Trick #7:
  You can do anything you want to the equation, As long as you Preserve Equality:
  This means: The left side of the equation is always equal to the right side of the equation.
  - So back to the example:
    We start with: E = I * R, where I is on the right side, but we want it on the left side.
    Remember from Trick #3 above:
    so we can switch around Ohm's law, and write it as: I * R = E
  - So now I is on the left side where we want it, and Equality is Preserved, the left side is still equal to the right side,
  - but now we have to somehow move R to the right side.
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- Trick #8:
  Equality is Preserved, if you: Do the same thing to both sides of the equation.
  - Multiply or Divide Both Sides by the same thing:
    - Remember what we're trying to do:
      We have an equation, and Just Looking at the Left Side, we have: I * R
      
      But we don't want the R there.
      We need I, our unknown variable, to be alone, by itself on the left side.
      So how can we get rid of the R ?
      
      If we divide by R, we'll get: (I * R) / R = I, because R / R = 1 and can be cancelled/ignored.
      This is the result we want, with our unknown I on the left side, by itself.
      
      But now to preserve equality, we must do the same things to both sides of the equation.
      If we divide the left side by R, we must also divide the right side by R
    - In this example, we'll Divide:
      Start with our equation: I * R = E
      If you divide both sides of the equation by R, you get:
      (I * R) / R = E / R
    - This can be written as: I * (R / R) = E / R
      which is: I * 1 = E / R
      which is: I = E / R
    - And this is the solution to the problem:
      We have an equation with the unknown variable I on the left side by itself,
      and known variables E and R on the right side,
      so we have everything we need to calculate the value of I:
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- Trick #9:
  Using Parentheses and Brackets:
  - Parentheses, Brackets, and Braces are always used in pairs, two of the same kind.
    A pair of parentheses ( ) is equivalent to a pair of brackets [ ], which is equivalent to a pair of braces { } .
    The choice of which to use is arbitrary and personal preference, though parentheses are the most common.
  - Parentheses are sometimes used when they're not really needed, just for clarity, for example in a long and complicated expression.
  - However parentheses MUST be used:
  - For example: If you have an equation like: a = 4 + 2 * 3
    How do you know whether to perform the addition first, or the multiplication?
  - The default priority/order of mathematical operations
    The default priority order is (partly):
  - The full priority order is:
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  - But, sometimes it's good to use parentheses even if you don't need them:
    - In the example formula:
      a = 4 + 2 * 3
      The proper order of operation is clearly defined by the default priority rule,
      Do the multiplication first, then the addition.
    - Yet it's still a little confusing, especially to beginning algebra students.
      So this might be a time when you could use parentheses, even though they're not strictly required:
    - The answer is the same, especially considering the first priority, which is calculate inside parentheses first,
      but it surely is less confusing.
    - Another example:
      When we were juggling things above, we divided the left side of the equation by R:
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- powers and exponents
- units
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