What are Variables?
Variables are letters that are used to represent unknown numbers. They have two main roles:
- A placeholder for missing values that make a statement true.
- Their use in writing formulas.
Intuition: for missing values
"What is two plus four?"
To answer the question above we could write out " \[2 + 4 = ?\] " using a question mark for the missing answer because we do not know the answer.
If we were asked, "What number plus 3 equals 7?" We could use a question mark again and write:
\[? + 3 = 7\]
Using basic arithmetic we find that \(2 + 4 = 6\) and \(4 + 3 = 7\). In algebra instead of using question marks for missing numbers we use letters such as x, y, a, b, and c. We would write "\(2 + 4 = ?\)" as "\(2 + 4 = x\)". To solve we would perform something similar to this:
\[ \begin{align} 2 + 4 &= x \\ 6 &= x \end{align} \]
Notice that since 2 + 4 = 6 we re-write the problem under the original one and write 6 instead of "2 + 4". This is called "simplifying an equation". Algebra is based on solving problems by re-writing simpler versions of it. Since we are left with " 6 = x " we are finished. That is all there is to it.
When there are no more numbers to add we are done. What is left is one number on one side of the equal sign and one letter by itself on the other. If we have an answer like "6 = x" this is called "solving for x". To solve problems in algebra we often solve for x or another letter. We call the number we get "the solution for x".
How do we really know that 6 = x? What does that even mean?
To prove 6 = x we can use a method called substitution to check our work. To use substitution we "plug" the number we think is our answer in for x in the original equation:
Write the original equation
\[ 2 + 4 = x \]
Substitute your earlier solution for \(x\)
\[ 2 + 4 = 6 \]
Simplify all terms until there is no more work that can be done
\[ 6 = 6 \]
In the end we got " 6 = 6 " which is true. If done correctly you should be left we two numbers; one on the left hand side of the equation and one on the right. If both numbers match then the answer is correct. If they do not match then the answer is wrong. If you have plus or minus signs and more and only two numbers you still have more work to do to simplify the problem.
Intuition: for formulas
A formula is a generic equation that sets the rules for finding answers to a group of related math problems. Area problems for a rectangle use the formula "length x width", or "base times height":
"What is the area of a garden 4 feet by 8 feet?"
1. Write the formula for area of a rectangle (base x height = Area)
2. Substitute given values for base and height
3. Work the problem
4. Give final answer with units if they exist
\[ \begin{align} bh &= A\\ 4(8) &= A\\ 32 &= A\\ \end{align} \] The area of the garden is "32 square feet"
After solving equations do not forget to write the units of an answer. "Units" are what is being counted. For an area of 4 feet by 8 feet the unit label is "square feet" written as 32sqft. or 32ft2.
Here we use the letters b,h, and A as variables because they are abbreviations for words that represent distances.