One step solutions

We've studied using words and tables in our solutions to problems. Now we will learn and practice solving simple problem situations using algebra. By taking "one step" we will solve basic problems.

We will learn this technique of algebra by starting with simple problems that we could easily do in our heads. We will show the tie between our "intuitive" solution and the "algebraic" solution. Hopefully as the problems situations become more and more complicated, we will be able to lean strongly upon solid algebra techniques to pull us through.

Ok then, let's take a look at the "algebra" method.

Consider this work we've seen before:

A variable is an unknown or changing value. A variable expression is an expression that contains a variable. Example 1. Kudzu is an amazingly fast-growing plant. It can grow up to 12 inches in one day. It grows so fast that it can overtake an entire house! Days 1 2 3 4 ... d Length of Kudzu vine (inches) 12 24 36 48 ... 12d Note that the variable d stands for the number of days and the variable expression 12d means "12 times d". The variable expression had to be 12d because the length of Kudzu was always 12 times the day number.

Suppose the problem question for the Kudzu example was "How long would it take a Kudzu vine to grow 72 inches?"

Here's a sample solution:

"Intuitive solution"	"Hmm,...it grows 12 inches every day and we want it to grow 72 inches. So how many times will it grow a 12 inch block to total 72 inches? That's gonna be 72 ÷ 12 = 6. So it'll take 6 days."
"Algebra Solution"	Algebra Reasoning Let d = the number of days to grow 72 inches. We must define the variable. Since Kuzdu grows 12 inches per day and the total should be 72. Divide both sides by 12 to "undo" the multiply by 12. Simplifying

Although the algebra method may seem longer in this simple problem, we will first practice using algebra in "simple" situations. This way we will be able to compare the steps of algebra to the solution and steps we can do in our heads. This way we will emphasize the tie between simple algebra and common "intuition".

Let's look at a few parts of the algebra solution a little more closely.

Did our algebra solution have these parts critical parts every good solution will have?

• the problem situation explained: Notice when the variable was defined and the first equation was written, the situation was explained.
• clear, easy to follow reasoning: Notice the steps that were shown.
• the correct answer in correct units: Notice that the last line only states d = 6. This works since the variable was clearly defined.
  That is, when the solution says " d = 6" it really is saying "the number of days to grow 72 inches = 6" because that's exactly what d means according to the definition.

How is this any different than what we did in Unit 1?

• In Unit 1 we worked with variable expressions. Here we are using variable equations.
• In Unit 1 we wrote expressions. Here we are writing and solving equations.
• In Unit 1 we were describe a trend using variables. Here we are solving a specific situation using equations.

So what is an equation exactly? Head on over to this dictionary for one good definition.

Let's look at the process of solving with algebra.

Check out these videos for some good insight into the process:

• the basic background of algebra
• "undoing" addition to solve
• "undoing" subtraction to solve
• "undoing" multipliction to solve
• "undoing" division to solve
• Bonus: an engineer's perspective on the importance of algebra

To do your work using algebra can be difficult on a computer. Here are some helpful options.

• Use Microsoft Word's equation editor. A tutorial can be found here.
• Use a pencil on a printed paper. When finished, make a digital image of your work with a scanner or camera. Then attached the image(s) as needed.