Algebra Chapter 1 Section 4: Introduction to Problem Solving

Hello internet, and welcome to the Algebra Lecture Series from the Science of Life.  This entry is an introduction to algebraic problem solving.  In Algebra, there are five steps for problem solving, which I'll cover here.
  1. Familiarize yourself with the problem situation.  This is basically a situation of knowing what the context of the problem is.  Is the context taxes?  Grocery shopping?  Tipping?  Scaling up or down the ingredients of a cake to get a different size cake than what the recipe says the size will be?  Here are the sub-steps for this contexts step:
    1. Read the problem carefully.  Read it aloud if need be to understand the problem.
    2. List the information and state the question being asked.  Select variables to represent anything which is unknown and clearly state what the variables represent, and be descriptive about the statement.
    3. Obtain any relevant information and equations.  If you're painting a room, find the equation for finding wall space, taking into account that windows and doors are not walls, and find how many square feet one gallon of paint covers.
    4. Create a table if needed with all variables and known information.  Look for any possible patterns.
    5. Make and label a graph or drawing.  This has been covered in the Physics Lecture Series.  Note: When making a graph in two dimensions, the vertical axis is typically labeled as the dependent variable (the y-variable, up is positive) and the horizontal axis is typically labeled as the independent variable (the x-variable, right is positive).
    6. Estimate your answer and check.  For graphs, this means eye-ball what number corresponds with the answer and plug it into the appropriate equations.
  2. Translate the problem into mathematical language.  Some of the translations are tabulated in Section 1 of this lecture series.
  3. Carry out any mathematical manipulations required to get a variable alone on one side of the equation.  If the translation from step 2 is a single equation, than solve for the unknown.  It it's multiple equations, then solve the "n-equations-n-unknowns" model, which will be covered in Chapter 3.
  4. Check your answer with the initial problem to make sure that it makes intuitive sense and is mathematically valid.  Intuitive sense is as important as mathematically correct; after all, if you're looking for mass and you get a negative number, then you know you've done something wrong.
  5. State the answer clearly in both mathematical terms [either an equality/inequality (x=5 or x ≥ 5) or an interval (0 ≤ x ≤ 5)].
That's the end of this section.  If you have any questions, please leave them is the comments.  Like and share this post if you found it helpful.  And until next time, stay curious.

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