Algebra: Variables
Variables
Max earns #4# euros per hour.
The number of hours that Max works varies.
If he works #\blue 3# hours a day, he earns #4 \times \blue3 =12# euro.
On another day, he works #\blue 5# hours and earns #4 \times \blue 5=20# euro.
In math, a varying quantity is represented by a letter, such as #\blue x#.
Now assume Max works #\blue x# hours.
Max earns #4# euro per hour; hence, he earns #4\times \blue x# euro in total.
If Max works #\blue 7# hours, then #\blue x = \blue 7# and he will earn #4 \times \blue 7 =28# euro in total.
With #\blue 3,\blue5# or #\blue 7# hours of work, Max earns:
\[\begin{array}{rcl}
4 \times \blue3 &=& 12 \\
4 \times \blue 5 &=& 20 \\
4 \times \blue 7 &=& 28
\end{array}\]
With #\blue x# hours of work, Max earns:
\[\begin{array}{c}
4 \times \blue x
\end{array}\]
In the example above, we use mathematical symbols to express a situation algebraically. Here #4 \times \blue x# is called an algebraic expression and the letter #\blue x# is called a variable.
Algebra is about calculating with variables. On the right-hand side, you can see more examples of algebraic expressions. In those examples, #\blue x#, #\green y#, and #\purple z# are variables. In algebraic expressions, we can replace the variables with numbers. Calculating with variables is done in the same way as calculating with numbers.
Examples
\[\begin{array}{c}
\blue x+5 \\ \\
\sqrt{\green y-3} \\ \\
\purple z^2 + \purple z
\end{array}\]
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