What is a Variable?

What is a Variable? exercise

• Explain what a variable is.

  Hints

  For example, Paul is $12$ years old. The age of Paul's sister, Anne, is unknown. Together they are $21$ years old.

  This leads to the equation $12+a=21$, where $a$ is the variable representing Anne's age.

  Say Paul has two apples and five pears and both of them have the same price, $x$. If he pays $1.40$ dollars in total, then the corresponding equation is $2x+5x=1.4$.

  Because both summands have the factor $x$ in common, we can factor it to get $2x+5x=(2+5)x=7x$.

  An example of an algebraic equation is $2x-2=12$.

  Solution

  A variable is a symbol, often $x$, $y$, or any letter used to represent an unknown value. For instance,

  • $c$ for coyote
  • $s$ for snake

  This letter represents a value in equations and inequalities.

  Variables can be used in algebraic expressions, together with constants, operators, etc. Using variables can be useful when solving word problems.

  Variable can be treated in the same way as numbers, as they represent quantities:

  • We can add or subtract them $2x+3x=5x$ or $5s-3s=2s$.
  • We can multiply them $(2x)(3x)=6x^2$.
  • We can also divide them $(6x^2)\div(3x)=2x$.

• Find the right equation.

  Hints

  Not every variable must be $x$ or $y$. You can also use the first letter of the unknown value, for instance.

  Just look at the equation with the corresponding words.

  $\le$ or $\ge$ are used for inequalities.

  Solution

  Let's have a look at what we know:

  • Cello has $15$ steaks in total.
  • The number of coyotes is unknown. Let's use $c$ to represent the number of coyotes.
  • Because each coyote devours one steak, we have to subtract $c$ from $15$, $15-c$.
  • $5$ steaks need to be left for the campers to eat. So we must have that $15-c$ equals $5$ in the end.

  With this knowledge, we can establish the following equation: $15-c=5$.

• Find the words which can be represented by variables.

  Hints

  Keep in mind that a variable represents an unknown quantity, which would like to be known.

  Perhaps it is helpful to establish the equations which represent each word problem. For example, the first one is given by $(15)(x)=60$.

  Solution

  When solving a word problem, the first step is determining the unknown values which would like to be known, then assigning a variable to each of these unknown values.

  Together with the known values you can establish the corresponding equation.

  Lunch

  The unknown value is the number of new members. Each given number can't be a variable at all.

  Muffins

  In this problem the amount of flour for each muffin is wanted.

  Scrambled eggs

  You know the number of eggs per portion as well as the total number of eggs. The number of portions is wanted.

  Firewood

  Here we know the total number of collected wood. We also know that Mr. Bianca samples 20 pieces. Cello and Jessica collect the same number, which is unknown.

• Establish the equation for the given word problems.

  Hints

  Check the known values for each word problem. There is still one unknown value. This is the variable.

  First, try to establish the equation on your own. Just have a look at the following example:

  Jessica prepares a certain number of scrambled eggs portions. For one portion of scrambled eggs she uses $2$ eggs. In total she uses $30$ eggs. This leads to the equation $(2)(x)=30$.

  Equations can be written in different ways. Let's have a look at the following example, $a+3=5$.

  This equation can also be written as

  • $3+a=5$ using the commutative property,
  • $3=5-a$, or
  • $a=5-3$ using opposite operations.

  Solution

  In each of the given problems there is a wanted value. Each time the corresponding variable is $x$.

  Filling the Bus

  Because $52$ people are already sitting on the bus and in total $80$ people can fit on the bus, the number of people needed to fill the bus is given by the equation $52+x=80$.

  How Many Buses

  We want to know the number of buses. We assign $x$ to this number, and we get the equation $(40)(x)=80$.

  How Many People

  We have to add the number of males and females together to get the total number of people, or in other words, $28+52=x$.

  Apples

  Because each of the $28$ people have the same number of apples, i.e. $x$, and the total number is given by $84$, then we get the equation $(28)(x)=84$.

• Find the variable.

  Hints

  A variable is often a letter, like $x$, $y$, or any other letter, and is used to represent an unknown quantity.

  $+$ or $-$ are operators, while $=$ is a relation.

  Solution

  You can recognize a variable as it is usually a letter (you can also use symbols like a star or a smiley face, but maybe they aren't always the most convenient to use). Looking at our given equations, we can see that:

  • $x$, $y$ or $a$, $b$, $c$ are variables.
  • $10$, $5$ are numbers.
  • $+$ or $-$ are operators.
  • $=$ is a relation.

  So let's examine the following equations:

  • $5+\mathbf{x}=11$

  Here the variable is given by $x$.

  • $15-\mathbf{c}=5$

  The variable $c$ stands for the unknown number of coyotes.

  • $(4)(3)=\mathbf{s}$

  $s$ is the variable for the unknown number of rattle snakes.

• Write the equation.

  Hints

  Consider the unknown value and assign the variable $x$ to it.

  Next collect the known values, i.e. the number of kittens as well as the number of toys for each kitten.

  You can establish different equations using the commutative property. For example, $x+4=8$ or $4+x=8$.

  You can also write the equation above as $4=8-x$ or $x=8-4$.

  Solution

  Sue likes to know the total number of toys. Let's represent this unknown by $x$.

  Because she buys $2$ toys for each of her new kittens, $x$ is given by multiplying the given numbers

  $(3)(2)=x$.

  Sure, you can move the order of multiplication using the commutative property to get

  $(2)(3)=x$.

  You can also divide the equation above by $2$ or $3$ to get

  $2=x\div 3$ or $3=x\div 2$.

  However, the solution is always the same $x=6$. Sue, enjoy your toy shopping and your time with the new kittens.