Variable

A variable is a certain symbol like a Latin letter – $X$/$Y$/$Z$ that can change and take on a different value each time.
The meaning of the variable is to symbolize something for us – quantity, price, an element within a set, or some value.
The variable can appear in a function or as part of some expression and can also be called an unknown or a parameter.

A variable can be dependent or independent.
A dependent variable is one that depends on something – for example, $Y$ in a function, which will depend on the $X$ we substitute for it.
An independent variable is one that does not depend on anything - usually $X$ (it can have a certain range of values) and it will ultimately determine the value of the dependent variable.
Important to know –
Although the variable changes and can take on a different number each time, if the same variable appears several times in the same expression or function, it will be identical in all its occurrences in the function.

Let's see an example:

Given the following function:
$Y=X^2+2X+6$

• How many variables are there in the function?
• Which is the dependent variable and which is the independent variable?
• If $X=2$ what will be the value of the variable $Y$.

Solution:

1. In the given function, there are only $2$ variables: $X$ and $Y$.
   Note that $X$ appears twice and in different forms – once squared and once multiplied by $2$, but it is the same $X$ with the same value in both instances, so we will treat it as one variable.
2. The dependent variable in the function is $Y$ because its value depends on what we substitute for $X$.
   The independent variable is $X$ because it does not depend on any other parameter and can take any value.
3. To know the value of the variable $Y$ when $X=2$, we need to substitute $X=2$ in every instance of $X$ in the function. We get:
   $Y=2^2+2*2+6$
   $Y=4+4+6$
   $Y=12$

Sometimes, we encounter word problems where we are asked to find the value of something specific, like the price of a shirt, the number of slices of cake, or the number of children.

We can take the data from the question and turn it into an algebraic equation using variables.
When we have an algebraic equation composed of various variables based on the data in the question, we can easily solve the word problem.

Let's see an example:

Advanced word problem with variables –

Saar went to the nearby mall and decided to buy $2$ shirts, $1$ pair of pants, and $3$ packs of cards.
It is known that a shirt costs $X ₪$ and the price of the pants is $3$ times the price of the shirt.
It is also known that the price of a pack of cards is half the price of the pants.
Additionally, Saar paid $8$ dollars for parking.

1. Present the prices of all items using variables.
2. Find out the price of the pants and the price of the pack of cards if you know that $X=30$.
3. How much did Saar pay in total during his visit to the mall, based on item 2?

Solution:

1.    We will represent the data of the question using variables:
Price of a shirt = $X$ known according to the data of the question 

Price of pants =
According to the question data, it is known that the price of pants is $3$ times higher than the price of the shirt.

Price of a pack of cards = $3X \over2$
If the price of the pants is $3X$, and the price of a pack of cards is half the price of the pants, then the price of the pack of cards is $3X$ divided by $2$.

2.    If $X=30$
The price of the pants is $3*30=90$
$90$
And the price of a pack of cards is:
$90:2=45$
3.     To know how much Sa'ar paid in total at the mall, we need to build an equation:
Note that it is given that:
Sa'ar decided to buy $2$ shirts, $1$ pair of pants, and $3$ packs of cards and paid an additional $8$ for parking.

Therefore, the equation will be:

$2*30+90+3*45+8=$
$60+90+135+8=293$

Saar spent $293$ in total during the entire visit to the mall, including parking.